Copyright (c) 2002 Association of Mizar Users
environ
vocabulary FUNCT_1, BOOLE, ABSVALUE, EUCLID, PRE_TOPC, SQUARE_1, RELAT_1,
SUBSET_1, ARYTM_3, METRIC_1, RCOMP_1, FUNCT_5, TOPMETR, COMPTS_1,
JGRAPH_4, ORDINAL2, TOPS_2, ARYTM_1, COMPLEX1, MCART_1, PCOMPS_1,
JGRAPH_3, BORSUK_1, TOPREAL1, TOPREAL2, JORDAN3, PSCOMP_1, REALSET1,
JORDAN5C, JORDAN6, ARYTM, SEQ_1;
notation ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, XBOOLE_0, ABSVALUE,
EUCLID, TARSKI, RELAT_1, TOPS_2, FUNCT_1, FUNCT_2, NAT_1, STRUCT_0,
TOPMETR, PCOMPS_1, COMPTS_1, METRIC_1, SQUARE_1, RCOMP_1, PSCOMP_1,
BINOP_1, PRE_TOPC, JGRAPH_1, JGRAPH_3, TOPREAL1, JORDAN5C, JORDAN6,
TOPREAL2, JGRAPH_4, GRCAT_1;
constructors REAL_1, ABSVALUE, TOPREAL1, TOPS_2, RCOMP_1, PSCOMP_1, TOPREAL2,
WELLFND1, JGRAPH_3, JORDAN5C, JORDAN6, JGRAPH_4, GRCAT_1, BORSUK_3,
TOPRNS_1;
clusters XREAL_0, STRUCT_0, RELSET_1, FUNCT_1, EUCLID, PRE_TOPC, TOPMETR,
SQUARE_1, PSCOMP_1, BORSUK_1, METRIC_1, BORSUK_2, BORSUK_3, MEMBERED;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
definitions TARSKI, JORDAN6;
theorems TARSKI, XBOOLE_0, XBOOLE_1, AXIOMS, RELAT_1, FUNCT_1, FUNCT_2,
TOPS_1, TOPS_2, PRE_TOPC, TOPMETR, JORDAN6, EUCLID, REAL_1, REAL_2,
JGRAPH_1, SEQ_4, SQUARE_1, PSCOMP_1, METRIC_1, JGRAPH_2, RCOMP_1,
COMPTS_1, RFUNCT_2, SETWISEO, BORSUK_1, TOPREAL1, TOPREAL3, TOPREAL5,
JGRAPH_3, ABSVALUE, COMPLEX1, JORDAN5A, JORDAN5B, JORDAN7, HEINE,
JGRAPH_4, PCOMPS_1, JORDAN5C, JORDAN1B, XREAL_0, TREAL_1, GRCAT_1,
TSEP_1, JORDAN1A, JORDAN1, TOPRNS_1, XCMPLX_0, XCMPLX_1;
schemes FUNCT_2, JGRAPH_2;
begin :: Preliminaries
reserve x,a for real number;
theorem Th1:
a>=0 & (x-a)*(x+a)>=0 implies -a>=x or x>=a
proof assume a>=0 & (x-a)*(x+a)>=0;
then x-a>=0 & x+a>=0 or x-a<=0 & x+a<=0 by SQUARE_1:25;
then x-a+a>=0+a or x+a<=0 by REAL_1:55;
then x-(a-a)>=0+a or x+a<=0 by XCMPLX_1:37;
then x>=0+a or x+a-a<=0-a by REAL_1:49,XCMPLX_1:17;
then x>=0+a or x+(a-a)<=0-a by XCMPLX_1:29;
then x>=a or x<=0-a by XCMPLX_1:25;
hence -a>=x or x>=a by XCMPLX_1:150;
end;
theorem Th2: a<=0 & x<a implies x^2>a^2
proof assume A1: a<=0 & x<a;
then --a<=0;
then A2: -a>=0 by REAL_1:66;
-x>-a by A1,REAL_1:50;
then (-x)^2>(-a)^2 by A2,SQUARE_1:78;
then x^2>(-a)^2 by SQUARE_1:61;
hence thesis by SQUARE_1:61;
end;
theorem Th3: for p being Point of TOP-REAL 2 st |.p.|<=1
holds -1<=p`1 & p`1<=1 & -1<=p`2 & p`2<=1
proof let p be Point of TOP-REAL 2;
assume A1: |.p.|<=1;
set a=|.p.|;
A2: a>=0 by TOPRNS_1:26;
A3: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:10;
then a^2-(p`1)^2=(p`2)^2 by XCMPLX_1:26;
then a^2-(p`1)^2>=0 by SQUARE_1:72;
then a^2-(p`1)^2+(p`1)^2>=0+(p`1)^2 by REAL_1:55;
then a^2>=(p`1)^2 by XCMPLX_1:27;
then A4: -a<=p`1 & p`1<=a by A2,JGRAPH_2:5;
a^2-(p`2)^2=(p`1)^2 by A3,XCMPLX_1:26;
then a^2-(p`2)^2>=0 by SQUARE_1:72;
then a^2-(p`2)^2+(p`2)^2>=0+(p`2)^2 by REAL_1:55;
then a^2>=(p`2)^2 by XCMPLX_1:27;
then A5: -a<=p`2 & p`2<=a by A2,JGRAPH_2:5;
-a>=-1 by A1,REAL_1:50;
hence thesis by A1,A4,A5,AXIOMS:22;
end;
theorem Th4: for p being Point of TOP-REAL 2 st |.p.|<=1 & p`1<>0 & p`2<>0
holds -1<p`1 & p`1<1 & -1<p`2 & p`2<1
proof let p be Point of TOP-REAL 2;
assume A1: |.p.|<=1 & p`1<>0 & p`2<>0;
set a=|.p.|;
A2: a>=0 by TOPRNS_1:26;
A3: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:10;
then a^2-(p`1)^2=(p`2)^2 by XCMPLX_1:26;
then a^2-(p`1)^2>0 by A1,SQUARE_1:74;
then a^2-(p`1)^2+(p`1)^2>0+(p`1)^2 by REAL_1:67;
then a^2>(p`1)^2 by XCMPLX_1:27;
then A4: -a<p`1 & p`1<a by A2,JGRAPH_2:6;
a^2-(p`2)^2=(p`1)^2 by A3,XCMPLX_1:26;
then a^2-(p`2)^2>0 by A1,SQUARE_1:74;
then a^2-(p`2)^2+(p`2)^2>0+(p`2)^2 by REAL_1:67;
then a^2>(p`2)^2 by XCMPLX_1:27;
then A5: -a<p`2 & p`2<a by A2,JGRAPH_2:6;
-a>=-1 by A1,REAL_1:50;
hence thesis by A1,A4,A5,AXIOMS:22;
end;
theorem for a,b,d,e,r3 being Real,PM,PM2 being non empty MetrStruct,
x being Element of PM,
x2 being Element of PM2
st d<=a & a<=b & b<=e
& PM=Closed-Interval-MSpace(a,b)
& PM2=Closed-Interval-MSpace(d,e)
& x=x2 & x in the carrier of PM & x2 in the carrier of PM2
holds Ball(x,r3) c= Ball(x2,r3)
proof let a,b,d,e,r3 be Real,PM,PM2 be non empty MetrStruct,
x be Element of PM,
x2 be Element of PM2;
assume A1: d<=a & a<=b & b<=e
& PM=Closed-Interval-MSpace(a,b)
& PM2=Closed-Interval-MSpace(d,e) &
x=x2 & x in the carrier of PM & x2 in the carrier of PM2;
then A2: d<=b by AXIOMS:22;
then A3: d<=e by A1,AXIOMS:22;
A4: a<=e by A1,AXIOMS:22;
let z be set;assume z in Ball(x,r3);
then z in {y where y is Element of PM: dist(x,y) < r3 }
by METRIC_1:18;
then consider y being Element of PM such that
A5: y=z & dist(x,y)<r3;
A6: the carrier of PM=[.a,b.] by A1,TOPMETR:14;
A7: a in [.d,e.] by A1,A4,TOPREAL5:1;
b in [.d,e.] by A1,A2,TOPREAL5:1;
then A8: [.a,b.] c= [.d,e.] by A7,RCOMP_1:16;
A9: (the distance of PM).(x,y)
= real_dist.(x,y) by A1,METRIC_1:def 14,TOPMETR:def 1;
A10: dist(x,y)= (the distance of PM).(x,y) by METRIC_1:def 1;
y in [.a,b.] by A6;
then reconsider y3=y as Element of PM2 by A1,A3,A8,TOPMETR:14
;
real_dist.(x,y)=
(the distance of PM2).(x2,y3) by A1,METRIC_1:def 14,TOPMETR:def 1
.=dist(x2,y3) by METRIC_1:def 1;
then z in {y2 where y2 is Element of PM2:
dist(x2,y2)<r3} by A5,A9,A10;
hence thesis by METRIC_1:18;
end;
theorem Th6: for a,b,d,e being real number,
B being Subset of Closed-Interval-TSpace(d,e)
st d<=a & a<=b & b<=e & B=[.a,b.] holds
Closed-Interval-TSpace(a,b)=Closed-Interval-TSpace(d,e)|B
proof let a,b,d,e be real number,
B be Subset of Closed-Interval-TSpace(d,e);
assume A1: d<=a & a<=b & b<=e & B=[.a,b.];
then A2: d<=b by AXIOMS:22;
then A3: d<=e by A1,AXIOMS:22;
A4: a<=e by A1,AXIOMS:22;
reconsider A=[.d,e.] as non empty Subset of R^1
by A1,A2,TOPMETR:24,TOPREAL5:1;
reconsider B2=[.a,b.] as non empty Subset of R^1
by A1,TOPMETR:24,TOPREAL5:1;
A5: a in [.d,e.] by A1,A4,TOPREAL5:1;
b in [.d,e.] by A1,A2,TOPREAL5:1;
then A6: [.a,b.] c= [.d,e.] by A5,RCOMP_1:16;
A7: Closed-Interval-TSpace(a,b)=R^1|B2 by A1,TOPMETR:26;
Closed-Interval-TSpace(d,e)=R^1|A by A3,TOPMETR:26;
hence thesis by A1,A6,A7,JORDAN6:47;
end;
theorem for a,b being real number, B being Subset of I[01]
st 0<=a & a<=b & b<=1 & B=[.a,b.] holds
Closed-Interval-TSpace(a,b)=I[01]|B by Th6,TOPMETR:27;
theorem Th8: for X being TopStruct,
Y,Z being non empty TopStruct,f being map of X,Y,
h being map of Y,Z st h is_homeomorphism & f is continuous
holds h*f is continuous
proof let X be TopStruct,Y,Z be non empty TopStruct,f be map of X,Y,
h be map of Y,Z;
assume A1: h is_homeomorphism &
f is continuous;
then h is continuous by TOPS_2:def 5;
hence h*f is continuous by A1,TOPS_2:58;
end;
theorem Th9: for X,Y,Z being TopStruct, f being map of X,Y,
h being map of Y,Z st h is_homeomorphism & f is one-to-one
holds h*f is one-to-one
proof let X,Y,Z be TopStruct, f be map of X,Y, h be map of Y,Z;
assume A1: h is_homeomorphism & f is one-to-one;
then h is one-to-one by TOPS_2:def 5;
hence h*f is one-to-one by A1,FUNCT_1:46;
end;
theorem Th10: for X being TopStruct,S,V being non empty TopStruct,
B being non empty Subset of S,f being map of X,S|B, g being map of S,V,
h being map of X,V
st h=g*f & f is continuous & g is continuous holds h is continuous
proof let X be TopStruct,S,V be non empty TopStruct,
B be non empty Subset of S,
f be map of X,S|B, g be map of S,V,
h being map of X,V;
assume that A1: h=g*f & f is continuous and A2: g is continuous;
now let P be Subset of V;
A3: (g*f)"P = f"(g"P) by RELAT_1:181;
now assume P is closed;
then A4: g"P is closed by A2,PRE_TOPC:def 12;
A5: [#](S|B)=B by PRE_TOPC:def 10;
A6: the carrier of S|B =B by JORDAN1:1;
then B /\ g"P c= the carrier of S|B by XBOOLE_1:17;
then reconsider F=B /\ g"P as Subset of S|B;
A7: F is closed by A4,A5,PRE_TOPC:43;
A8: rng f /\ (the carrier of S|B)= rng f by XBOOLE_1:28;
h"P=f"(rng f /\ g"P) by A1,A3,RELAT_1:168
.=f"(rng f /\ ((the carrier of S|B) /\ g"P)) by A8,XBOOLE_1:16
.=f"F by A6,RELAT_1:168;
hence h"P is closed by A1,A7,PRE_TOPC:def 12;
end;
hence P is closed implies h"P is closed;
end;
hence thesis by PRE_TOPC:def 12;
end;
theorem Th11:for a,b,d,e,s1,s2,t1,t2 being Real,h being map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e) st
h is_homeomorphism & h.s1=t1 & h.s2=t2 & h.a=d & h.b=e & d<=e &
t1<=t2 & s1 in [.a,b.] & s2 in [.a,b.] holds s1<=s2
proof let a,b,d,e,s1,s2,t1,t2 be Real,h be map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e);
assume A1: h is_homeomorphism & h.s1=t1 & h.s2=t2 & h.a=d & h.b=e & d<=e
& t1<=t2 & s1 in [.a,b.] & s2 in [.a,b.];
then A2: h is one-to-one by TOPS_2:def 5;
A3: a<=s2 & s2<=b by A1,TOPREAL5:1;
A4: a<=s1 & s1<=b by A1,TOPREAL5:1;
then A5: a<=b by AXIOMS:22;
A6: dom h=[#](Closed-Interval-TSpace(a,b)) by A1,TOPS_2:def 5
.=the carrier of Closed-Interval-TSpace(a,b) by PRE_TOPC:12
.=[.a,b.] by A5,TOPMETR:25;
A7: h is continuous by A1,TOPS_2:def 5;
A8: the carrier of Closed-Interval-TSpace(a,b)
=[.a,b.] by A5,TOPMETR:25;
A9: the carrier of Closed-Interval-TSpace(d,e)
=[.d,e.] by A1,TOPMETR:25;
A10: h is one-to-one by A1,TOPS_2:def 5;
[.s2,s1.] c= the carrier of Closed-Interval-TSpace(a,b)
by A3,A4,A8,TREAL_1:1;
then reconsider B=[.s2,s1.] as Subset of Closed-Interval-TSpace(a,b)
;
reconsider Bb=[.s2,s1.] as Subset of
Closed-Interval-TSpace(a,b) by A3,A4,A8,TREAL_1:1;
assume A11: s1>s2;
reconsider f3=h|Bb as map of Closed-Interval-TSpace(a,b)|B,
Closed-Interval-TSpace(d,e) by JGRAPH_3:12;
A12: f3 is continuous by A7,TOPMETR:10;
reconsider C=[.d,e.] as non empty Subset of R^1
by A1,TOPMETR:24,TOPREAL5:1;
A13: R^1|C=Closed-Interval-TSpace(d,e) by A1,TOPMETR:26;
A14: Closed-Interval-TSpace(s2,s1)
=Closed-Interval-TSpace(a,b)|B by A3,A4,A11,Th6;
then f3 is map of Closed-Interval-TSpace(s2,s1),R^1
by A13,JORDAN6:4;
then reconsider f=h|B as map of Closed-Interval-TSpace(s2,s1),R^1;
dom f=the carrier of Closed-Interval-TSpace(s2,s1)
by FUNCT_2:def 1;
then A15: dom f=[.s2,s1.] by A11,TOPMETR:25;
A16: f is continuous by A12,A13,A14,JORDAN6:4;
set t=(t1+t2)/2;
s2 in B by A11,TOPREAL5:1;
then A17: f.s2=t2 by A1,FUNCT_1:72;
s1 in B by A11,TOPREAL5:1;
then A18: f.s1=t1 by A1,FUNCT_1:72;
t1<>t2 by A1,A2,A6,A11,FUNCT_1:def 8;
then A19: t1<t2 by A1,REAL_1:def 5;
then t1+t1<t1+t2 by REAL_1:67;
then (t1+t1)/2<(t1+t2)/2 by REAL_1:73;
then A20: (2*t1)/2<t by XCMPLX_1:11;
t1+t2<t2+t2 by A19,REAL_1:67;
then (t1+t2)/2<(t2+t2)/2 by REAL_1:73;
then (2*t2)/2>t by XCMPLX_1:11;
then A21: t2>t & t>t1 by A20,XCMPLX_1:90;
then consider r being Real such that
A22: f.r =t & s2<r & r <s1 by A11,A16,A17,A18,TOPREAL5:13;
A23: r<b by A4,A22,AXIOMS:22;
a<r by A3,A22,AXIOMS:22;
then A24: r in [.a,b.] by A23,TOPREAL5:1;
[.s1,b.] c= the carrier of Closed-Interval-TSpace(a,b)
by A4,A8,TREAL_1:1;
then reconsider B1=[.s1,b.] as Subset of Closed-Interval-TSpace(a,b)
;
reconsider B1b=[.s1,b.] as Subset of
Closed-Interval-TSpace(a,b) by A4,A8,TREAL_1:1;
reconsider f4=h|B1b as map of Closed-Interval-TSpace(a,b)|B1,
Closed-Interval-TSpace(d,e) by JGRAPH_3:12;
A25: Closed-Interval-TSpace(s1,b)
=Closed-Interval-TSpace(a,b)|B1 by A4,Th6;
A26: f4 is continuous by A7,TOPMETR:10;
f4 is map of Closed-Interval-TSpace(s1,b),R^1
by A13,A25,JORDAN6:4;
then reconsider f1=h|B1 as map of Closed-Interval-TSpace(s1,b),R^1;
A27: f1 is continuous by A13,A25,A26,JORDAN6:4;
s2 in dom f by A11,A15,TOPREAL5:1;
then t2 in rng f3 by A17,FUNCT_1:def 5;
then A28: d<=t2 & t2<=e by A9,TOPREAL5:1;
then A29: s1<b by A1,A4,A19,REAL_1:def 5;
A30: s1 in B1 by A4,TOPREAL5:1;
A31: b in B1 by A4,TOPREAL5:1;
A32: f1.s1= t1 by A1,A30,FUNCT_1:72;
A33: f1.b= e by A1,A31,FUNCT_1:72;
e>t & t>t1 by A21,A28,AXIOMS:22;
then consider r1 being Real such that
A34: f1.r1 =t & s1<r1 & r1 <b by A27,A29,A32,A33,TOPREAL5:12;
a<r1 by A4,A34,AXIOMS:22;
then A35: r1 in [.a,b.] by A34,TOPREAL5:1;
A36: r1 in B1 by A34,TOPREAL5:1;
r in [.s2,s1.] by A22,TOPREAL5:1;
then h.r = t by A22,FUNCT_1:72 .=h.r1 by A34,A36,FUNCT_1:72;
hence contradiction by A6,A10,A22,A24,A34,A35,FUNCT_1:def 8;
end;
theorem Th12:for a,b,d,e,s1,s2,t1,t2 being Real,h being map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e) st
h is_homeomorphism & h.s1=t1 & h.s2=t2 & h.a=e & h.b=d & e>=d &
t1>=t2 & s1 in [.a,b.] & s2 in [.a,b.] holds s1<=s2
proof let a,b,d,e,s1,s2,t1,t2 be Real,h be map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e);
assume A1: h is_homeomorphism & h.s1=t1 & h.s2=t2 & h.a=e & h.b=d & e>=d
& t1>=t2 & s1 in [.a,b.] & s2 in [.a,b.];
then A2: h is one-to-one by TOPS_2:def 5;
A3: a<=s2 & s2<=b by A1,TOPREAL5:1;
A4: a<=s1 & s1<=b by A1,TOPREAL5:1;
then A5: a<=b by AXIOMS:22;
A6: dom h=[#](Closed-Interval-TSpace(a,b)) by A1,TOPS_2:def 5
.=the carrier of Closed-Interval-TSpace(a,b) by PRE_TOPC:12
.=[.a,b.] by A5,TOPMETR:25;
A7: h is continuous by A1,TOPS_2:def 5;
A8: the carrier of Closed-Interval-TSpace(a,b)
=[.a,b.] by A5,TOPMETR:25;
A9: the carrier of Closed-Interval-TSpace(d,e)
=[.d,e.] by A1,TOPMETR:25;
A10: h is one-to-one by A1,TOPS_2:def 5;
[.s2,s1.] c= the carrier of Closed-Interval-TSpace(a,b)
by A3,A4,A8,TREAL_1:1;
then reconsider B=[.s2,s1.] as Subset of Closed-Interval-TSpace(a,b)
;
reconsider Bb=[.s2,s1.] as Subset of
Closed-Interval-TSpace(a,b) by A3,A4,A8,TREAL_1:1;
assume A11: s1>s2;
reconsider f3=h|Bb as map of Closed-Interval-TSpace(a,b)|B,
Closed-Interval-TSpace(d,e) by JGRAPH_3:12;
A12: f3 is continuous by A7,TOPMETR:10;
reconsider C=[.d,e.] as non empty Subset of R^1
by A1,TOPMETR:24,TOPREAL5:1;
A13: R^1|C=Closed-Interval-TSpace(d,e) by A1,TOPMETR:26;
A14: Closed-Interval-TSpace(s2,s1)
=Closed-Interval-TSpace(a,b)|B by A3,A4,A11,Th6;
then f3 is map of Closed-Interval-TSpace(s2,s1),R^1
by A13,JORDAN6:4;
then reconsider f=h|B as map of Closed-Interval-TSpace(s2,s1),R^1;
dom f=the carrier of Closed-Interval-TSpace(s2,s1)
by FUNCT_2:def 1;
then A15: dom f=[.s2,s1.] by A11,TOPMETR:25;
A16: f is continuous by A12,A13,A14,JORDAN6:4;
set t=(t1+t2)/2;
s2 in B by A11,TOPREAL5:1;
then A17: f.s2=t2 by A1,FUNCT_1:72;
s1 in B by A11,TOPREAL5:1;
then A18: f.s1=t1 by A1,FUNCT_1:72;
t1<>t2 by A1,A2,A6,A11,FUNCT_1:def 8;
then A19: t1>t2 by A1,REAL_1:def 5;
then t1+t1>t1+t2 by REAL_1:67;
then (t1+t1)/2>(t1+t2)/2 by REAL_1:73;
then A20: (2*t1)/2>t by XCMPLX_1:11;
t1+t2>t2+t2 by A19,REAL_1:67;
then (t1+t2)/2>(t2+t2)/2 by REAL_1:73;
then (2*t2)/2<t by XCMPLX_1:11;
then A21: t2<t & t<t1 by A20,XCMPLX_1:90;
then consider r being Real such that
A22: f.r =t & s2<r & r <s1 by A11,A16,A17,A18,TOPREAL5:12;
A23: r<b by A4,A22,AXIOMS:22;
a<r by A3,A22,AXIOMS:22;
then A24: r in [.a,b.] by A23,TOPREAL5:1;
[.s1,b.] c= the carrier of Closed-Interval-TSpace(a,b)
by A4,A8,TREAL_1:1;
then reconsider B1=[.s1,b.] as Subset of Closed-Interval-TSpace(a,b)
;
reconsider B1b=[.s1,b.] as Subset of
Closed-Interval-TSpace(a,b) by A4,A8,TREAL_1:1;
reconsider f4=h|B1b as map of Closed-Interval-TSpace(a,b)|B1,
Closed-Interval-TSpace(d,e) by JGRAPH_3:12;
A25: Closed-Interval-TSpace(s1,b)
=Closed-Interval-TSpace(a,b)|B1 by A4,Th6;
A26: f4 is continuous by A7,TOPMETR:10;
f4 is map of Closed-Interval-TSpace(s1,b),R^1
by A13,A25,JORDAN6:4;
then reconsider f1=h|B1 as map of Closed-Interval-TSpace(s1,b),R^1;
A27: f1 is continuous by A13,A25,A26,JORDAN6:4;
s2 in dom f by A11,A15,TOPREAL5:1;
then t2 in rng f3 by A17,FUNCT_1:def 5;
then A28: d<=t2 & t2<=e by A9,TOPREAL5:1;
then A29: s1<b by A1,A4,A19,REAL_1:def 5;
A30: s1 in B1 by A4,TOPREAL5:1;
A31: b in B1 by A4,TOPREAL5:1;
A32: f1.s1= t1 by A1,A30,FUNCT_1:72;
A33: f1.b= d by A1,A31,FUNCT_1:72;
d<t & t<t1 by A21,A28,AXIOMS:22;
then consider r1 being Real such that
A34: f1.r1 =t & s1<r1 & r1 <b by A27,A29,A32,A33,TOPREAL5:13;
a<r1 by A4,A34,AXIOMS:22;
then A35: r1 in [.a,b.] by A34,TOPREAL5:1;
A36: r1 in B1 by A34,TOPREAL5:1;
r in [.s2,s1.] by A22,TOPREAL5:1;
then h.r= t by A22,FUNCT_1:72 .=h.r1 by A34,A36,FUNCT_1:72;
hence contradiction by A6,A10,A22,A24,A34,A35,FUNCT_1:def 8;
end;
theorem for n being Nat holds
-(0.REAL n)=0.REAL n
proof let n be Nat;
0.REAL n+0.REAL n=0.REAL n by EUCLID:31;
hence thesis by EUCLID:41;
end;
begin :: Fashoda Meet Theorems for Circle in Special Case
theorem Th14:
for f,g being map of I[01],TOP-REAL 2,a,b,c,d being Real,
O,I being Point of I[01] st O=0 & I=1 &
f is continuous one-to-one &
g is continuous one-to-one & a <> b & c <> d &
(f.O)`1=a & (c <=(f.O)`2 & (f.O)`2 <=d) &
(f.I)`1=b & (c <=(f.I)`2 & (f.I)`2 <=d) &
(g.O)`2=c & (a <=(g.O)`1 & (g.O)`1 <=b) &
(g.I)`2=d & (a <=(g.I)`1 & (g.I)`1 <=b) &
(for r being Point of I[01] holds
(a >=(f.r)`1 or (f.r)`1>=b or c >=(f.r)`2 or (f.r)`2>=d) &
(a >=(g.r)`1 or (g.r)`1>=b or c >=(g.r)`2 or (g.r)`2>=d))
holds rng f meets rng g
proof let f,g be map of I[01],TOP-REAL 2,a,b,c,d be Real,
O,I be Point of I[01];
assume A1: O=0 & I=1 &
f is continuous one-to-one & g is continuous one-to-one & a <> b & c <> d &
(f.O)`1=a & (c <=(f.O)`2 & (f.O)`2 <=d) &
(f.I)`1=b & (c <=(f.I)`2 & (f.I)`2 <=d) &
(g.O)`2=c & (a <=(g.O)`1 & (g.O)`1 <=b) &
(g.I)`2=d & (a <=(g.I)`1 & (g.I)`1 <=b) &
(for r being Point of I[01] holds
(a >=(f.r)`1 or (f.r)`1>=b or c >=(f.r)`2 or (f.r)`2>=d) &
(a >=(g.r)`1 or (g.r)`1>=b or c >=(g.r)`2 or (g.r)`2>=d));
then A2: a <= b by AXIOMS:22;
c <= d by A1,AXIOMS:22;
then a < b & c < d by A1,A2,REAL_1:def 5;
hence thesis by A1,JGRAPH_2:55;
end;
Lm1: 0 in [.0,1.] & 1 in [.0,1.] by RCOMP_1:15;
theorem Th15: for f being map of I[01],TOP-REAL 2 st
f is continuous one-to-one
ex f2 being map of I[01],TOP-REAL 2 st f2.0=f.1 & f2.1=f.0 &
rng f2=rng f & f2 is continuous & f2 is one-to-one
proof let f be map of I[01],TOP-REAL 2;
assume A1: f is continuous one-to-one;
A2: I[01] is compact by HEINE:11,TOPMETR:27;
A3: dom f=the carrier of I[01] by FUNCT_2:def 1;
then A4: f.1 in rng f by Lm1,BORSUK_1:83,FUNCT_1:12;
reconsider P=rng f as non empty Subset of TOP-REAL 2 by A3,Lm1,BORSUK_1:83,
FUNCT_1:12;
consider f1 being map of I[01],(TOP-REAL 2)|P such that
A5: f1=f & f1 is_homeomorphism by A1,A2,JGRAPH_1:64;
f.0 in rng f by A3,Lm1,BORSUK_1:83,FUNCT_1:12;
then reconsider p1=f.0,p2=f.1 as Point of TOP-REAL 2 by A4;
P is_an_arc_of p1,p2 by A5,TOPREAL1:def 2;
then P is_an_arc_of p2,p1 by JORDAN5B:14;
then consider f3 being map of I[01], (TOP-REAL 2)|P such that
A6: f3 is_homeomorphism & f3.0 = p2 & f3.1 = p1 by TOPREAL1:def 2;
A7: rng f3=[#]((TOP-REAL 2)|P) by A6,TOPS_2:def 5
.=P by PRE_TOPC:def 10;
consider f4 being map of I[01], (TOP-REAL 2) such that
A8: f3=f4 & f4 is continuous & f4 is one-to-one by A6,JORDAN7:15;
thus thesis by A6,A7,A8;
end;
reserve p,q for Point of TOP-REAL 2;
theorem Th16:
for f,g being map of I[01],TOP-REAL 2,
C0,KXP,KXN,KYP,KYN being Subset of TOP-REAL 2,
O,I being Point of I[01] st O=0 & I=1 &
f is continuous one-to-one &
g is continuous one-to-one &
C0={p: |.p.|<=1}&
KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} &
KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} &
KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} &
KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} &
f.O in KXN & f.I in KXP & g.O in KYP & g.I in KYN &
rng f c= C0 & rng g c= C0 holds rng f meets rng g
proof let f,g be map of I[01],TOP-REAL 2,
C0,KXP,KXN,KYP,KYN be Subset of TOP-REAL 2,
O,I be Point of I[01];
assume A1: O=0 & I=1 &
f is continuous one-to-one &
g is continuous one-to-one &
C0={p: |.p.|<=1}&
KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} &
KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} &
KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} &
KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} &
f.O in KXN & f.I in KXP & g.O in KYP & g.I in KYN &
rng f c= C0 & rng g c= C0;
then consider g2 being map of I[01],TOP-REAL 2 such that
A2: g2.0=g.1 & g2.1=g.0 &
rng g2=rng g & g2 is continuous one-to-one by Th15;
thus thesis by A1,A2,JGRAPH_3:55;
end;
theorem Th17: for f,g being map of I[01],TOP-REAL 2,
C0,KXP,KXN,KYP,KYN being Subset of TOP-REAL 2,
O,I being Point of I[01] st O=0 & I=1 &
f is continuous one-to-one &
g is continuous one-to-one &
C0={p: |.p.|>=1}&
KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} &
KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} &
KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} &
KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} &
f.O in KXN & f.I in KXP & g.O in KYN & g.I in KYP &
rng f c= C0 & rng g c= C0 holds rng f meets rng g
proof let f,g be map of I[01],TOP-REAL 2,
C0,KXP,KXN,KYP,KYN be Subset of TOP-REAL 2,
O,I be Point of I[01];
assume A1: O=0 & I=1 &
f is continuous one-to-one & g is continuous one-to-one &
C0 = {p: |.p.|>=1}&
KXP = {q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} &
KXN = {q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} &
KYP = {q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} &
KYN = {q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} &
f.O in KXN & f.I in KXP & g.O in KYN & g.I in KYP &
rng f c= C0 & rng g c= C0;
Sq_Circ"*f is Function of the carrier of I[01],
the carrier of TOP-REAL 2 by FUNCT_2:19,JGRAPH_3:39;
then reconsider ff=Sq_Circ"*f as map of I[01],TOP-REAL 2;
Sq_Circ"*g is Function of the carrier of I[01],
the carrier of TOP-REAL 2 by FUNCT_2:19,JGRAPH_3:39;
then reconsider gg=Sq_Circ"*g as map of I[01],TOP-REAL 2;
consider h1 being map of (TOP-REAL 2),(TOP-REAL 2)
such that A2:h1=(Sq_Circ") & h1 is continuous by JGRAPH_3:52;
A3:dom ff=the carrier of I[01] by FUNCT_2:def 1;
A4:dom gg=the carrier of I[01] by FUNCT_2:def 1;
A5:dom f=the carrier of I[01] by FUNCT_2:def 1;
A6:dom g=the carrier of I[01] by FUNCT_2:def 1;
A7:ff is continuous by A1,A2,TOPS_2:58;
A8: Sq_Circ" is one-to-one by FUNCT_1:62,JGRAPH_3:32;
then A9:ff is one-to-one by A1,FUNCT_1:46;
A10:gg is continuous by A1,A2,TOPS_2:58;
A11:gg is one-to-one by A1,A8,FUNCT_1:46;
A12: (ff.O)`1=-1 & (ff.I)`1=1 & (gg.O)`2=-1 & (gg.I)`2=1
proof
A13: (ff.O)=(Sq_Circ").(f.O) by A3,FUNCT_1:22;
consider p1 being Point of TOP-REAL 2 such that
A14: f.O=p1 &( |.p1.|=1 & p1`2>=p1`1 & p1`2<=-p1`1) by A1;
A15:p1<>0.REAL 2 &
(p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2<=-p1`1)
by A14,TOPRNS_1:24;
then A16:Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2/p1`1)^2
)]|
by JGRAPH_3:38;
reconsider px=ff.O as Point of TOP-REAL 2;
set q=px;
A17: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) &
px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2) by A13,A14,A16,EUCLID:56;
(p1`2/p1`1)^2 >=0 by SQUARE_1:72;
then 1+(p1`2/p1`1)^2>=1+0 by REAL_1:55;
then 1+(p1`2/p1`1)^2>0 by AXIOMS:22;
then A18:sqrt(1+(p1`2/p1`1)^2)>0 by SQUARE_1:93;
A19:now assume
A20: px`1=0 & px`2=0;
then A21:p1`1=0 by A17,A18,XCMPLX_1:6;
p1`2=0 by A17,A18,A20,XCMPLX_1:6;
hence contradiction by A15,A21,EUCLID:57,58;
end;
p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 &
p1`2*sqrt(1+(p1`2/p1`1)^2) <= (-p1`1)*sqrt(1+(p1`2/p1`1)^2)
by A14,A18,AXIOMS:25;
then p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2/p1`1
)^2)
or px`2>=px`1 & px`2<=-px`1 by A17,A18,AXIOMS:25,XCMPLX_1:175;
then A22:px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1
by A17,A18,AXIOMS:25,XCMPLX_1:175;
A23:p1=Sq_Circ.px by A13,A14,FUNCT_1:54,JGRAPH_3:32,54;
A24:Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|
by A19,A22,JGRAPH_2:11,JGRAPH_3:def 1;
A25: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1
= q`1/sqrt(1+(q`2/q`1)^2) &
(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2
= q`2/sqrt(1+(q`2/q`1)^2) by EUCLID:56;
(q`2/q`1)^2 >=0 by SQUARE_1:72;
then 1+(q`2/q`1)^2>=1+0 by REAL_1:55;
then A26:1+(q`2/q`1)^2>0 by AXIOMS:22;
then A27:sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:93;
px`1<>0 by A19,A22;
then A28: (px`1)^2<>0 by SQUARE_1:73;
now
(|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by A23,A24,A25,JGRAPH_3:10
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by A26,SQUARE_1:def 4
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A26,SQUARE_1:def 4
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)
by XCMPLX_1:63;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)=(1)*(1+(q`2/q`1
)^2)
by A14,SQUARE_1:59;
then ((q`1)^2+(q`2)^2)=(1+(q`2/q`1)^2) by A26,XCMPLX_1:88;
then (px`1)^2+(px`2)^2=1+(px`2)^2/(px`1)^2 by SQUARE_1:69;
then (px`1)^2+(px`2)^2-1=(px`2)^2/(px`1)^2 by XCMPLX_1:26;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2=(px`2)^2 by A28,XCMPLX_1:88;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2=(px`2)^2 by XCMPLX_1:29;
then (px`1)^2*(px`1)^2+((px`2)^2-1)*(px`1)^2=(px`2)^2 by XCMPLX_1:8;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2=0 by XCMPLX_1:14
;
then (px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2=0
by XCMPLX_1:40;
then (px`1)^2*(px`1)^2+(px`1)^2*(px`2)^2-(px`1)^2*1-(px`2)^2=0
by XCMPLX_1:29;
then (px`1)^2*(px`1)^2-(px`1)^2*1+(px`1)^2*(px`2)^2-(1)*(px`2)^2=0
by XCMPLX_1:29;
then (px`1)^2*((px`1)^2-1)+(px`1)^2*(px`2)^2-(1)*(px`2)^2=0
by XCMPLX_1:40;
then (px`1)^2*((px`1)^2-1)+((px`1)^2*(px`2)^2-(1)*(px`2)^2)=0
by XCMPLX_1:29;
hence ((px`1)^2-1)*(px`1)^2+((px`1)^2-1)*(px`2)^2=0 by XCMPLX_1:40;
end;
then A29:((px`1)^2-1)*((px`1)^2+(px`2)^2)=0 by XCMPLX_1:8;
((px`1)^2+(px`2)^2)<>0 by A19,COMPLEX1:2;
then (px`1)^2-1=0 by A29,XCMPLX_1:6;
then (px`1-1)*(px`1+1)=0 by SQUARE_1:59,67;
then A30: px`1-1=0 or px`1+1=0 by XCMPLX_1:6;
A31: now assume A32: px`1=1;
then A33:p1`1>0 by A23,A24,A25,A27,REAL_2:127;
-p1`2>=--p1`1 by A14,REAL_1:50;
then -p1`2>0 by A23,A24,A25,A27,A32,REAL_2:127;
then --p1`2<-0 by REAL_1:50;
hence contradiction by A14,A33,AXIOMS:22;
end;
A34: (ff.I)=(Sq_Circ").(f.I) by A3,FUNCT_1:22;
consider p2 being Point of TOP-REAL 2 such that
A35: f.I=p2 & (|.p2.|=1 & p2`2<=p2`1 & p2`2>=-p2`1) by A1;
A36:p2<>0.REAL 2 &
(p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2<=-p2`1)
by A35,TOPRNS_1:24;
then A37:Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2/p2`1)^2
)]|
by JGRAPH_3:38;
reconsider py=ff.I as Point of TOP-REAL 2;
A38: py`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) &
py`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A34,A35,A37,EUCLID:56;
(p2`2/p2`1)^2 >=0 by SQUARE_1:72;
then 1+(p2`2/p2`1)^2>=1+0 by REAL_1:55;
then 1+(p2`2/p2`1)^2>0 by AXIOMS:22;
then A39:sqrt(1+(p2`2/p2`1)^2)>0 by SQUARE_1:93;
A40:now assume
A41: py`1=0 & py`2=0;
then A42:p2`1=0 by A38,A39,XCMPLX_1:6;
p2`2=0 by A38,A39,A41,XCMPLX_1:6;
hence contradiction by A36,A42,EUCLID:57,58;
end;
A43: now
p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/p2`1)^2)
or py`2>=py`1 & py`2<=-py`1 by A35,A39,AXIOMS:25;
hence
p2`2*sqrt(1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -py`1<=py
`2
or py`2>=py`1 & py`2<=-py`1 by A38,A39,AXIOMS:25,XCMPLX_1:175;
end;
A44:p2=Sq_Circ.py by A34,A35,FUNCT_1:54,JGRAPH_3:32,54;
A45:Sq_Circ.py=|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|
by A38,A40,A43,JGRAPH_2:11,JGRAPH_3:def 1;
A46: (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`1
= py`1/sqrt(1+(py`2/py`1)^2) &
(|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`2
= py`2/sqrt(1+(py`2/py`1)^2) by EUCLID:56;
(py`2/py`1)^2 >=0 by SQUARE_1:72;
then 1+(py`2/py`1)^2>=1+0 by REAL_1:55;
then A47:1+(py`2/py`1)^2>0 by AXIOMS:22;
then A48:sqrt(1+(py`2/py`1)^2)>0 by SQUARE_1:93;
py`1<>0 by A38,A40,A43;
then A49: (py`1)^2<>0 by SQUARE_1:73;
now
(|.p2.|)^2= (py`1/sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
by A44,A45,A46,JGRAPH_3:10
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
by SQUARE_1:69
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2
+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2
by SQUARE_1:69
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2
by A47,SQUARE_1:def 4
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(1+(py`2/py`1)^2)
by A47,SQUARE_1:def 4
.= ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2) by XCMPLX_1:63;
then ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2)*(1+(py`2/py`1)^2)
=(1)*(1+(py`2/py`1)^2) by A35,SQUARE_1:59;
then ((py`1)^2+(py`2)^2)=(1+(py`2/py`1)^2)
by A47,XCMPLX_1:88;
then (py`1)^2+(py`2)^2=1+(py`2)^2/(py`1)^2 by SQUARE_1:69;
then (py`1)^2+(py`2)^2-1=(py`2)^2/(py`1)^2 by XCMPLX_1:26;
then ((py`1)^2+(py`2)^2-1)*(py`1)^2=(py`2)^2 by A49,XCMPLX_1:88;
then ((py`1)^2+((py`2)^2-1))*(py`1)^2=(py`2)^2 by XCMPLX_1:29;
then (py`1)^2*(py`1)^2+((py`2)^2-1)*(py`1)^2=(py`2)^2 by XCMPLX_1:8;
then (py`1)^2*(py`1)^2+(py`1)^2*((py`2)^2-1)-(py`2)^2=0 by XCMPLX_1:14
;
then (py`1)^2*(py`1)^2+((py`1)^2*(py`2)^2-(py`1)^2*1)-(py`2)^2=0
by XCMPLX_1:40;
then (py`1)^2*(py`1)^2+(py`1)^2*(py`2)^2-(py`1)^2*1-(py`2)^2=0
by XCMPLX_1:29;
then (py`1)^2*(py`1)^2-(py`1)^2*1+(py`1)^2*(py`2)^2-(1)*(py`2)^2=0
by XCMPLX_1:29;
then (py`1)^2*((py`1)^2-1)+(py`1)^2*(py`2)^2-(1)*(py`2)^2=0
by XCMPLX_1:40;
then (py`1)^2*((py`1)^2-1)+((py`1)^2*(py`2)^2-(1)*(py`2)^2)=0
by XCMPLX_1:29;
hence ((py`1)^2-1)*(py`1)^2+((py`1)^2-1)*(py`2)^2=0 by XCMPLX_1:40;
end;
then A50:((py`1)^2-1)*((py`1)^2+(py`2)^2)=0 by XCMPLX_1:8;
((py`1)^2+(py`2)^2)<>0 by A40,COMPLEX1:2;
then ((py`1)^2-1)=0 by A50,XCMPLX_1:6;
then (py`1-1)*(py`1+1)=0 by SQUARE_1:59,67;
then A51: py`1-1=0 or py`1+1=0 by XCMPLX_1:6;
A52: now assume A53: py`1=-1;
then A54:p2`1<0 by A44,A45,A46,A48,REAL_2:128;
-p2`2<=--p2`1 by A35,REAL_1:50;
then -p2`2<0 by A44,A45,A46,A48,A53,REAL_2:128;
then --p2`2>-0 by REAL_1:50;
hence contradiction by A35,A54,AXIOMS:22;
end;
A55: (gg.O)=(Sq_Circ").(g.O) by A4,FUNCT_1:22;
consider p3 being Point of TOP-REAL 2 such that
A56: g.O=p3 &( |.p3.|=1 & p3`2<=p3`1 & p3`2<=-p3`1) by A1;
A57: -p3`2>=--p3`1 by A56,REAL_1:50;
then A58:p3<>0.REAL 2 &
(p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2)
by A56,TOPRNS_1:24;
then A59:Sq_Circ".p3=|[p3`1*sqrt(1+(p3`1/p3`2)^2),p3`2*sqrt(1+(p3`1/p3`2)^2)
]|
by JGRAPH_3:40;
reconsider pz=gg.O as Point of TOP-REAL 2;
A60: pz`2 = p3`2*sqrt(1+(p3`1/p3`2)^2) &
pz`1 = p3`1*sqrt(1+(p3`1/p3`2)^2) by A55,A56,A59,EUCLID:56;
(p3`1/p3`2)^2 >=0 by SQUARE_1:72;
then 1+(p3`1/p3`2)^2>=1+0 by REAL_1:55;
then 1+(p3`1/p3`2)^2>0 by AXIOMS:22;
then A61:sqrt(1+(p3`1/p3`2)^2)>0 by SQUARE_1:93;
A62:now assume
A63: pz`2=0 & pz`1=0;
then A64:p3`2=0 by A60,A61,XCMPLX_1:6;
p3`1=0 by A60,A61,A63,XCMPLX_1:6;
hence contradiction by A58,A64,EUCLID:57,58;
end;
A65: now
p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 &
p3`1*sqrt(1+(p3`1/p3`2)^2) <= (-p3`2)*sqrt(1+(p3`1/p3`2)^2)
by A56,A57,A61,AXIOMS:25;
then p3`1<=p3`2 & (-p3`2)*sqrt(1+(p3`1/p3`2)^2) <= p3`1*sqrt(1+(p3`1/p3`2
)^2)
or pz`1>=pz`2 & pz`1<=-pz`2 by A60,A61,AXIOMS:25,XCMPLX_1:175;
hence
p3`1*sqrt(1+(p3`1/p3`2)^2) <= p3`2*sqrt(1+(p3`1/p3`2)^2) & -pz`2<=pz`1
or pz`1>=pz`2 & pz`1<=-pz`2 by A60,A61,AXIOMS:25,XCMPLX_1:175;
end;
A66:p3=Sq_Circ.pz by A55,A56,FUNCT_1:54,JGRAPH_3:32,54;
A67:Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
by A60,A62,A65,JGRAPH_2:11,JGRAPH_3:14;
A68: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2
= pz`2/sqrt(1+(pz`1/pz`2)^2) &
(|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1
= pz`1/sqrt(1+(pz`1/pz`2)^2) by EUCLID:56;
(pz`1/pz`2)^2 >=0 by SQUARE_1:72;
then 1+(pz`1/pz`2)^2>=1+0 by REAL_1:55;
then A69:1+(pz`1/pz`2)^2>0 by AXIOMS:22;
then A70:sqrt(1+(pz`1/pz`2)^2)>0 by SQUARE_1:93;
pz`2<>0 by A60,A62,A65;
then A71: (pz`2)^2<>0 by SQUARE_1:73;
A72:(|.p3.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by A66,A67,A68,JGRAPH_3:10
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2
+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A69,SQUARE_1:def 4
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2)
by A69,SQUARE_1:def 4
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:63;
now
((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)=(1)*(1+(pz`1/pz`2)^2
)
by A56,A72,SQUARE_1:59;
then ((pz`2)^2+(pz`1)^2)=(1+(pz`1/pz`2)^2) by A69,XCMPLX_1:88;
then (pz`2)^2+(pz`1)^2=1+(pz`1)^2/(pz`2)^2 by SQUARE_1:69;
then (pz`2)^2+(pz`1)^2-1=(pz`1)^2/(pz`2)^2 by XCMPLX_1:26;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2=(pz`1)^2 by A71,XCMPLX_1:88;
then ((pz`2)^2+((pz`1)^2-1))*(pz`2)^2=(pz`1)^2 by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2+((pz`1)^2-1)*(pz`2)^2=(pz`1)^2 by XCMPLX_1:8;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2=0 by XCMPLX_1:14
;
then (pz`2)^2*(pz`2)^2+((pz`2)^2*(pz`1)^2-(pz`2)^2*1)-(pz`1)^2=0
by XCMPLX_1:40;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*(pz`1)^2-(pz`2)^2*1-(pz`1)^2=0
by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2-(pz`2)^2*1+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2=0
by XCMPLX_1:29;hence
(pz`2)^2*((pz`2)^2-1)+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2=0
by XCMPLX_1:40;
end;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2*(pz`1)^2-(1)*(pz`1)^2)=0
by XCMPLX_1:29;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2-1)*(pz`1)^2=0 by XCMPLX_1:40;
then A73:((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)=0 by XCMPLX_1:8;
((pz`2)^2+(pz`1)^2)<>0 by A62,COMPLEX1:2;
then ((pz`2)^2-1)=0 by A73,XCMPLX_1:6;
then (pz`2-1)*(pz`2+1)=0 by SQUARE_1:59,67;
then pz`2-1=0 or pz`2+1=0 by XCMPLX_1:6;
then pz`2=0+1 or pz`2+1=0 by XCMPLX_1:27;
then A74: pz`2=1 or pz`2=0-1 by XCMPLX_1:26;
A75: now assume A76: pz`2=1;
then A77:p3`2>0 by A66,A67,A68,A70,REAL_2:127;
-p3`1>0 by A56,A66,A67,A68,A70,A76,REAL_2:127;
then --p3`1<-0 by REAL_1:50;
hence contradiction by A56,A77,AXIOMS:22;
end;
A78: (gg.I)=(Sq_Circ").(g.I) by A4,FUNCT_1:22;
consider p4 being Point of TOP-REAL 2 such that
A79: g.I=p4 &(
|.p4.|=1 & p4`2>=p4`1 & p4`2>=-p4`1) by A1;
A80: -p4`2<=--p4`1 by A79,REAL_1:50;
then A81:p4<>0.REAL 2 & (p4`1<=p4`2 & -p4`2<=p4`1 or p4`1>=p4`2 & p4`1<=-p4
`2)
by A79,TOPRNS_1:24;
then A82:Sq_Circ".p4=|[p4`1*sqrt(1+(p4`1/p4`2)^2),p4`2*sqrt(1+(p4`1/p4`2)^2
)]|
by JGRAPH_3:40;
reconsider pu=gg.I as Point of TOP-REAL 2;
A83: pu`2 = p4`2*sqrt(1+(p4`1/p4`2)^2) &
pu`1 = p4`1*sqrt(1+(p4`1/p4`2)^2) by A78,A79,A82,EUCLID:56;
(p4`1/p4`2)^2 >=0 by SQUARE_1:72;
then 1+(p4`1/p4`2)^2>=1+0 by REAL_1:55;
then 1+(p4`1/p4`2)^2>0 by AXIOMS:22;
then A84:sqrt(1+(p4`1/p4`2)^2)>0 by SQUARE_1:93;
A85:now assume
A86: pu`2=0 & pu`1=0;
then A87:p4`2=0 by A83,A84,XCMPLX_1:6;
p4`1=0 by A83,A84,A86,XCMPLX_1:6;
hence contradiction by A81,A87,EUCLID:57,58;
end;
A88: now
p4`1<=p4`2 & (-p4`2)*sqrt(1+(p4`1/p4`2)^2) <= p4`1*sqrt(1+(p4`1/p4`2)^2)
or pu`1>=pu`2 & pu`1<=-pu`2 by A79,A80,A84,AXIOMS:25;hence
p4`1*sqrt(1+(p4`1/p4`2)^2) <= p4`2*sqrt(1+(p4`1/p4`2)^2) & -pu`2<=pu`1
or pu`1>=pu`2 & pu`1<=-pu`2 by A83,A84,AXIOMS:25,XCMPLX_1:175;
end;
A89:p4=Sq_Circ.pu by A78,A79,FUNCT_1:54,JGRAPH_3:32,54;
A90:Sq_Circ.pu=|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|
by A83,A85,A88,JGRAPH_2:11,JGRAPH_3:14;
A91: (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`2
= pu`2/sqrt(1+(pu`1/pu`2)^2) &
(|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`1
= pu`1/sqrt(1+(pu`1/pu`2)^2) by EUCLID:56;
(pu`1/pu`2)^2 >=0 by SQUARE_1:72;
then 1+(pu`1/pu`2)^2>=1+0 by REAL_1:55;
then A92:1+(pu`1/pu`2)^2>0 by AXIOMS:22;
then A93:sqrt(1+(pu`1/pu`2)^2)>0 by SQUARE_1:93;
pu`2<>0 by A83,A85,A88;
then A94: (pu`2)^2<>0 by SQUARE_1:73;
now
(|.p4.|)^2= (pu`2/sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
by A89,A90,A91,JGRAPH_3:10
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
by SQUARE_1:69
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2
+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2
by SQUARE_1:69
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2
by A92,SQUARE_1:def 4
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(1+(pu`1/pu`2)^2)
by A92,SQUARE_1:def 4
.= ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2) by XCMPLX_1:63;
then ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2)*(1+(pu`1/pu`2)^2)=(1)*(1+(pu`1/pu
`2)^2)
by A79,SQUARE_1:59;
then ((pu`2)^2+(pu`1)^2)=(1+(pu`1/pu`2)^2)
by A92,XCMPLX_1:88;
then (pu`2)^2+(pu`1)^2=1+(pu`1)^2/(pu`2)^2 by SQUARE_1:69;
then (pu`2)^2+(pu`1)^2-1=(pu`1)^2/(pu`2)^2 by XCMPLX_1:26;
then ((pu`2)^2+(pu`1)^2-1)*(pu`2)^2=(pu`1)^2
by A94,XCMPLX_1:88;
then ((pu`2)^2+((pu`1)^2-1))*(pu`2)^2=(pu`1)^2 by XCMPLX_1:29;
then (pu`2)^2*(pu`2)^2+((pu`1)^2-1)*(pu`2)^2=(pu`1)^2 by XCMPLX_1:8;
then (pu`2)^2*(pu`2)^2+(pu`2)^2*((pu`1)^2-1)-(pu`1)^2=0 by XCMPLX_1:14
;
then (pu`2)^2*(pu`2)^2+((pu`2)^2*(pu`1)^2-(pu`2)^2*1)-(pu`1)^2=0
by XCMPLX_1:40;
then (pu`2)^2*(pu`2)^2+(pu`2)^2*(pu`1)^2-(pu`2)^2*1-(pu`1)^2=0
by XCMPLX_1:29;
then (pu`2)^2*(pu`2)^2-(pu`2)^2*1+(pu`2)^2*(pu`1)^2-(1)*(pu`1)^2=0
by XCMPLX_1:29;
then (pu`2)^2*((pu`2)^2-1)+(pu`2)^2*(pu`1)^2-(1)*(pu`1)^2=0
by XCMPLX_1:40;
then (pu`2)^2*((pu`2)^2-1)+((pu`2)^2*(pu`1)^2-(1)*(pu`1)^2)=0
by XCMPLX_1:29;
hence ((pu`2)^2-1)*(pu`2)^2+((pu`2)^2-1)*(pu`1)^2=0 by XCMPLX_1:40;
end;
then A95:((pu`2)^2-1)*((pu`2)^2+(pu`1)^2)=0 by XCMPLX_1:8;
((pu`2)^2+(pu`1)^2)<>0 by A85,COMPLEX1:2;
then ((pu`2)^2-1)=0 by A95,XCMPLX_1:6;
then (pu`2-1)*(pu`2+1)=0 by SQUARE_1:59,67;
then pu`2-1=0 or pu`2+1=0 by XCMPLX_1:6;
then A96: pu`2=0+1 or pu`2+1=0 by XCMPLX_1:27;
now assume A97: pu`2=-1;
then A98:p4`2<0 by A89,A90,A91,A93,REAL_2:128;
-p4`1<0 by A79,A89,A90,A91,A93,A97,REAL_2:128;
then --p4`1>-0 by REAL_1:50;
hence contradiction by A79,A98,AXIOMS:22;
end;
hence thesis by A30,A31,A51,A52,A74,A75,A96,XCMPLX_1:26,27;
end;
A99: for r being Point of I[01] holds
(-1>=(ff.r)`1 or (ff.r)`1>=1 or -1 >=(ff.r)`2 or (ff.r)`2>=1)
& (-1>=(gg.r)`1 or (gg.r)`1>=1 or -1 >=(gg.r)`2 or (gg.r)`2>=1)
proof let r be Point of I[01];
A100: (ff.r)=(Sq_Circ").(f.r) by A3,FUNCT_1:22;
f.r in rng f by A5,FUNCT_1:12;
then f.r in C0 by A1;
then consider p1 being Point of TOP-REAL 2 such that
A101: f.r=p1 & |.p1.|>=1 by A1;
A102:now per cases;
case A103: p1=0.REAL 2;
|.0.REAL 2.|=0 by TOPRNS_1:24;
hence contradiction by A101,A103;
case
A104:p1<>0.REAL 2 &
(p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2<=-p1`1);
then A105:Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2/p1`1)^2
)]|
by JGRAPH_3:38;
reconsider px=ff.r as Point of TOP-REAL 2;
set q=px;
A106: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) &
px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2) by A100,A101,A105,EUCLID:56;
(p1`2/p1`1)^2 >=0 by SQUARE_1:72;
then 1+(p1`2/p1`1)^2>=1+0 by REAL_1:55;
then 1+(p1`2/p1`1)^2>0 by AXIOMS:22;
then A107:sqrt(1+(p1`2/p1`1)^2)>0 by SQUARE_1:93;
A108:now assume
A109: px`1=0 & px`2=0;
then A110:p1`1=0 by A106,A107,XCMPLX_1:6;
p1`2=0 by A106,A107,A109,XCMPLX_1:6;
hence contradiction by A104,A110,EUCLID:57,58;
end;
A111: now
p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 &
p1`2*sqrt(1+(p1`2/p1`1)^2) <= (-p1`1)*sqrt(1+(p1`2/p1`1)^2)
by A104,A107,AXIOMS:25;
then p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2/p1`1
)^2)
or px`2>=px`1 & px`2<=-px`1 by A106,A107,AXIOMS:25,XCMPLX_1:175;
hence
p1`2*sqrt(1+(p1`2/p1`1)^2) <= p1`1*sqrt(1+(p1`2/p1`1)^2) & -px`1<=px`2
or px`2>=px`1 & px`2<=-px`1 by A106,A107,AXIOMS:25,XCMPLX_1:175;
end;
A112:p1=Sq_Circ.px by A100,A101,FUNCT_1:54,JGRAPH_3:32,54;
A113:Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|
by A106,A108,A111,JGRAPH_2:11,JGRAPH_3:def 1;
A114: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1
= q`1/sqrt(1+(q`2/q`1)^2) &
(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2
= q`2/sqrt(1+(q`2/q`1)^2) by EUCLID:56;
A115:(px`1)^2 >=0 by SQUARE_1:72;
(q`2/q`1)^2 >=0 by SQUARE_1:72;
then 1+(q`2/q`1)^2>=1+0 by REAL_1:55;
then A116:1+(q`2/q`1)^2>0 by AXIOMS:22;
(|.p1.|)^2>=|.p1.| by A101,JGRAPH_2:2;
then A117: (|.p1.|)^2>=1 by A101,AXIOMS:22;
px`1<>0 by A106,A108,A111;
then A118: (px`1)^2<>0 by SQUARE_1:73;
now
(|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by A112,A113,A114,JGRAPH_3:10
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by A116,SQUARE_1:def 4
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A116,SQUARE_1:def 4
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)
by XCMPLX_1:63;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)>=(1)*(1+(q`2/q
`1)^2)
by A116,A117,AXIOMS:25;
then ((q`1)^2+(q`2)^2)>=(1+(q`2/q`1)^2)
by A116,XCMPLX_1:88;
then (px`1)^2+(px`2)^2>=1+(px`2)^2/(px`1)^2 by SQUARE_1:69;
then (px`1)^2+(px`2)^2-1>=1+(px`2)^2/(px`1)^2-1 by REAL_1:49;
then (px`1)^2+(px`2)^2-1>=(px`2)^2/(px`1)^2 by XCMPLX_1:26;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>=(px`2)^2/(px`1)^2*(px`1)^2
by A115,AXIOMS:25;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>=(px`2)^2
by A118,XCMPLX_1:88;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2>=(px`2)^2 by XCMPLX_1:29;
then (px`1)^2*(px`1)^2+((px`2)^2-1)*(px`1)^2>=(px`2)^2 by XCMPLX_1:8;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2>=(px`2)^2-(px`2
)^2
by REAL_1:49;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2>=0 by XCMPLX_1:
14
;
then (px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2>=0
by XCMPLX_1:40;
then (px`1)^2*(px`1)^2+(px`1)^2*(px`2)^2-(px`1)^2*1-(px`2)^2>=0
by XCMPLX_1:29;
then (px`1)^2*(px`1)^2-(px`1)^2*1+(px`1)^2*(px`2)^2-(1)*(px`2)^2>=0
by XCMPLX_1:29;
then (px`1)^2*((px`1)^2-1)+(px`1)^2*(px`2)^2-(1)*(px`2)^2>=0
by XCMPLX_1:40;
then (px`1)^2*((px`1)^2-1)+((px`1)^2*(px`2)^2-(1)*(px`2)^2)>=0
by XCMPLX_1:29;hence
((px`1)^2-1)*(px`1)^2+((px`1)^2-1)*(px`2)^2>=0 by XCMPLX_1:40;
end;
then A119:((px`1)^2-1)*((px`1)^2+(px`2)^2)>=0 by XCMPLX_1:8;
A120: ((px`1)^2+(px`2)^2)<>0 by A108,COMPLEX1:2;
(px`2)^2>=0 by SQUARE_1:72;
then ((px`1)^2+(px`2)^2)>=0+0 by A115,REAL_1:55;
then ((px`1)^2-1)>=0 by A119,A120,SQUARE_1:25;
then (px`1-1)*(px`1+1)>=0 by SQUARE_1:59,67;
hence -1>=(ff.r)`1 or (ff.r)`1>=1 or
-1 >=(ff.r)`2 or (ff.r)`2>=1 by Th1;
case
A121:p1<>0.REAL 2 &
not(p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2<=-p1`1);
then A122:Sq_Circ".p1=|[p1`1*sqrt(1+(p1`1/p1`2)^2),p1`2*sqrt(1+(p1`1/p1`2)
^2)]|
by JGRAPH_3:38;
A123:p1<>0.REAL 2 &
(p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 & p1`1<=-p1`2)
by A121,JGRAPH_2:23;
reconsider pz=ff.r as Point of TOP-REAL 2;
A124: pz`2 = p1`2*sqrt(1+(p1`1/p1`2)^2) &
pz`1 = p1`1*sqrt(1+(p1`1/p1`2)^2) by A100,A101,A122,EUCLID:56;
(p1`1/p1`2)^2 >=0 by SQUARE_1:72;
then 1+(p1`1/p1`2)^2>=1+0 by REAL_1:55;
then 1+(p1`1/p1`2)^2>0 by AXIOMS:22;
then A125:sqrt(1+(p1`1/p1`2)^2)>0 by SQUARE_1:93;
A126:now assume
A127: pz`2=0 & pz`1=0;
then p1`2=0 by A124,A125,XCMPLX_1:6;
hence contradiction by A121,A124,A125,A127,XCMPLX_1:6;
end;
A128: now
p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 &
p1`1*sqrt(1+(p1`1/p1`2)^2) <= (-p1`2)*sqrt(1+(p1`1/p1`2)^2)
by A123,A125,AXIOMS:25;
then p1`1<=p1`2 & (-p1`2)*sqrt(1+(p1`1/p1`2)^2) <= p1`1*sqrt(1+(p1`1/p1`2
)^2)
or pz`1>=pz`2 & pz`1<=-pz`2 by A124,A125,AXIOMS:25,XCMPLX_1:175;
hence
p1`1*sqrt(1+(p1`1/p1`2)^2) <= p1`2*sqrt(1+(p1`1/p1`2)^2) & -pz`2<=pz`1
or pz`1>=pz`2 & pz`1<=-pz`2 by A124,A125,AXIOMS:25,XCMPLX_1:175;
end;
A129:p1=Sq_Circ.pz by A100,A101,FUNCT_1:54,JGRAPH_3:32,54;
A130:Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
by A124,A126,A128,JGRAPH_2:11,JGRAPH_3:14;
A131: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2
= pz`2/sqrt(1+(pz`1/pz`2)^2) &
(|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1
= pz`1/sqrt(1+(pz`1/pz`2)^2) by EUCLID:56;
A132:(pz`2)^2 >=0 by SQUARE_1:72;
(pz`1/pz`2)^2 >=0 by SQUARE_1:72;
then 1+(pz`1/pz`2)^2>=1+0 by REAL_1:55;
then A133:1+(pz`1/pz`2)^2>0 by AXIOMS:22;
(|.p1.|)^2>=|.p1.| by A101,JGRAPH_2:2;
then A134: (|.p1.|)^2>=1 by A101,AXIOMS:22;
pz`2<>0 by A124,A126,A128;
then A135: (pz`2)^2<>0 by SQUARE_1:73;
A136:(|.p1.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by A129,A130,A131,JGRAPH_3:10
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2
+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A133,SQUARE_1:def 4
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2)
by A133,SQUARE_1:def 4
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)
by XCMPLX_1:63;
now
((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)
>=(1)*(1+(pz`1/pz`2)^2)
by A133,A134,A136,AXIOMS:25;
then ((pz`2)^2+(pz`1)^2)>=(1+(pz`1/pz`2)^2)
by A133,XCMPLX_1:88;
then (pz`2)^2+(pz`1)^2>=1+(pz`1)^2/(pz`2)^2 by SQUARE_1:69;
then (pz`2)^2+(pz`1)^2-1>=1+(pz`1)^2/(pz`2)^2-1 by REAL_1:49;
then (pz`2)^2+(pz`1)^2-1>=(pz`1)^2/(pz`2)^2 by XCMPLX_1:26;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2>=(pz`1)^2/(pz`2)^2*(pz`2)^2
by A132,AXIOMS:25;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2>=(pz`1)^2
by A135,XCMPLX_1:88;
then ((pz`2)^2+((pz`1)^2-1))*(pz`2)^2>=(pz`1)^2 by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2+((pz`1)^2-1)*(pz`2)^2>=(pz`1)^2 by XCMPLX_1:8;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2
>=(pz`1)^2-(pz`1)^2 by REAL_1:49;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2>=0 by XCMPLX_1:
14
;
then (pz`2)^2*(pz`2)^2+((pz`2)^2*(pz`1)^2-(pz`2)^2*1)-(pz`1)^2>=0
by XCMPLX_1:40;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*(pz`1)^2-(pz`2)^2*1-(pz`1)^2>=0
by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2-(pz`2)^2*1+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2>=0
by XCMPLX_1:29;
hence (pz`2)^2*((pz`2)^2-1)+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2>=0
by XCMPLX_1:40;
end;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2*(pz`1)^2-(1)*(pz`1)^2)>=0
by XCMPLX_1:29;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2-1)*(pz`1)^2>=0
by XCMPLX_1:40;
then A137:((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)>=0 by XCMPLX_1:8;
A138: ((pz`2)^2+(pz`1)^2)<>0 by A126,COMPLEX1:2;
(pz`1)^2>=0 by SQUARE_1:72;
then (pz`2)^2+(pz`1)^2>=0+0 by A132,REAL_1:55;
then (pz`2)^2-1>=0 by A137,A138,SQUARE_1:25;
then (pz`2-1)*(pz`2+1)>=0 by SQUARE_1:59,67;
hence -1>=(ff.r)`1 or (ff.r)`1>=1 or
-1 >=(ff.r)`2 or (ff.r)`2>=1 by Th1;
end;
A139: (gg.r)=(Sq_Circ").(g.r) by A4,FUNCT_1:22;
g.r in rng g by A6,FUNCT_1:12;
then g.r in C0 by A1;
then consider p2 being Point of TOP-REAL 2 such that
A140: g.r=p2 & |.p2.|>=1 by A1;
now per cases;
case A141: p2=0.REAL 2;
|.0.REAL 2.|=0 by TOPRNS_1:24;
hence contradiction by A140,A141;
case
A142:p2<>0.REAL 2 &
(p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2<=-p2`1);
then A143:Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2/p2`1)^2
)]|
by JGRAPH_3:38;
reconsider px=gg.r as Point of TOP-REAL 2;
set q=px;
A144:Sq_Circ".p2=q by A4,A140,FUNCT_1:22;
A145: px`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) &
px`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A139,A140,A143,EUCLID:56;
(p2`2/p2`1)^2 >=0 by SQUARE_1:72;
then 1+(p2`2/p2`1)^2>=1+0 by REAL_1:55;
then 1+(p2`2/p2`1)^2>0 by AXIOMS:22;
then A146:sqrt(1+(p2`2/p2`1)^2)>0 by SQUARE_1:93;
A147:now assume
A148: px`1=0 & px`2=0;
then A149:p2`1=0 by A145,A146,XCMPLX_1:6;
p2`2=0 by A145,A146,A148,XCMPLX_1:6;
hence contradiction by A142,A149,EUCLID:57,58;
end;
A150: now
p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 &
p2`2*sqrt(1+(p2`2/p2`1)^2) <= (-p2`1)*sqrt(1+(p2`2/p2`1)^2)
by A142,A146,AXIOMS:25;
then p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/p2`1
)^2)
or px`2>=px`1 & px`2<=-px`1 by A145,A146,AXIOMS:25,XCMPLX_1:175
;hence
p2`2*sqrt(1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -px`1<=px
`2
or px`2>=px`1 & px`2<=-px`1 by A145,A146,AXIOMS:25,XCMPLX_1:175;
end;
A151:p2=Sq_Circ.px by A144,FUNCT_1:54,JGRAPH_3:32,54;
A152:Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|
by A145,A147,A150,JGRAPH_2:11,JGRAPH_3:def 1;
A153: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1
= q`1/sqrt(1+(q`2/q`1)^2) &
(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2
= q`2/sqrt(1+(q`2/q`1)^2) by EUCLID:56;
A154:(px`1)^2 >=0 by SQUARE_1:72;
(q`2/q`1)^2 >=0 by SQUARE_1:72;
then 1+(q`2/q`1)^2>=1+0 by REAL_1:55;
then A155:1+(q`2/q`1)^2>0 by AXIOMS:22;
(|.p2.|)^2>=|.p2.| by A140,JGRAPH_2:2;
then A156: (|.p2.|)^2>=1 by A140,AXIOMS:22;
px`1<>0 by A145,A147,A150;
then A157: (px`1)^2<>0 by SQUARE_1:73;
now
(|.p2.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by A151,A152,A153,JGRAPH_3:10
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by A155,SQUARE_1:def 4
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A155,SQUARE_1:def 4
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)
by XCMPLX_1:63;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)>=(1)*(1+(q`2/q
`1)^2)
by A155,A156,AXIOMS:25;
then ((q`1)^2+(q`2)^2)>=(1+(q`2/q`1)^2)
by A155,XCMPLX_1:88;
then (px`1)^2+(px`2)^2>=1+(px`2)^2/(px`1)^2 by SQUARE_1:69;
then (px`1)^2+(px`2)^2-1>=1+(px`2)^2/(px`1)^2-1 by REAL_1:49;
then (px`1)^2+(px`2)^2-1>=(px`2)^2/(px`1)^2 by XCMPLX_1:26;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>=(px`2)^2/(px`1)^2*(px`1)^2
by A154,AXIOMS:25;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>=(px`2)^2
by A157,XCMPLX_1:88;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2>=(px`2)^2 by XCMPLX_1:29;
then (px`1)^2*(px`1)^2+((px`2)^2-1)*(px`1)^2>=(px`2)^2 by XCMPLX_1:8;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2>=(px`2)^2-(px`2
)^2
by REAL_1:49;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2>=0 by XCMPLX_1:
14
;
then (px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2>=0
by XCMPLX_1:40;
then (px`1)^2*(px`1)^2+(px`1)^2*(px`2)^2-(px`1)^2*1-(px`2)^2>=0
by XCMPLX_1:29;
then (px`1)^2*(px`1)^2-(px`1)^2*1+(px`1)^2*(px`2)^2-(1)*(px`2)^2>=0
by XCMPLX_1:29;
then (px`1)^2*((px`1)^2-1)+(px`1)^2*(px`2)^2-(1)*(px`2)^2>=0
by XCMPLX_1:40;
then (px`1)^2*((px`1)^2-1)+((px`1)^2*(px`2)^2-(1)*(px`2)^2)>=0
by XCMPLX_1:29;hence
((px`1)^2-1)*(px`1)^2+((px`1)^2-1)*(px`2)^2>=0 by XCMPLX_1:40;
end;
then A158:((px`1)^2-1)*((px`1)^2+(px`2)^2)>=0 by XCMPLX_1:8;
A159: ((px`1)^2+(px`2)^2)<>0 by A147,COMPLEX1:2;
(px`2)^2>=0 by SQUARE_1:72;
then ((px`1)^2+(px`2)^2)>=0+0 by A154,REAL_1:55;
then ((px`1)^2-1)>=0 by A158,A159,SQUARE_1:25;
then (px`1-1)*(px`1+1)>=0 by SQUARE_1:59,67;
hence -1>=(gg.r)`1 or (gg.r)`1>=1 or -1 >=(gg.r)`2 or (gg.r)`2>=1
by Th1;
case
A160:p2<>0.REAL 2 &
not(p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2<=-p2`1);
then A161:Sq_Circ".p2=|[p2`1*sqrt(1+(p2`1/p2`2)^2),p2`2*sqrt(1+(p2`1/p2`2)^2
)]|
by JGRAPH_3:38;
A162:p2<>0.REAL 2 &
(p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 & p2`1<=-p2`2)
by A160,JGRAPH_2:23;
reconsider pz=gg.r as Point of TOP-REAL 2;
A163: pz`2 = p2`2*sqrt(1+(p2`1/p2`2)^2) &
pz`1 = p2`1*sqrt(1+(p2`1/p2`2)^2) by A139,A140,A161,EUCLID:56;
(p2`1/p2`2)^2 >=0 by SQUARE_1:72;
then 1+(p2`1/p2`2)^2>=1+0 by REAL_1:55;
then 1+(p2`1/p2`2)^2>0 by AXIOMS:22;
then A164:sqrt(1+(p2`1/p2`2)^2)>0 by SQUARE_1:93;
A165:now assume
A166: pz`2=0 & pz`1=0;
then p2`2=0 by A163,A164,XCMPLX_1:6;
hence contradiction by A160,A163,A164,A166,XCMPLX_1:6;
end;
A167: now
p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 &
p2`1*sqrt(1+(p2`1/p2`2)^2) <= (-p2`2)*sqrt(1+(p2`1/p2`2)^2)
by A162,A164,AXIOMS:25;
then p2`1<=p2`2 & (-p2`2)*sqrt(1+(p2`1/p2`2)^2) <= p2`1*sqrt(1+(p2`1/p2`2)
^2)
or pz`1>=pz`2 & pz`1<=-pz`2 by A163,A164,AXIOMS:25,XCMPLX_1:175;
hence
p2`1*sqrt(1+(p2`1/p2`2)^2) <= p2`2*sqrt(1+(p2`1/p2`2)^2) & -pz`2<=pz`1
or pz`1>=pz`2 & pz`1<=-pz`2 by A163,A164,AXIOMS:25,XCMPLX_1:175;
end;
A168:p2=Sq_Circ.pz by A139,A140,FUNCT_1:54,JGRAPH_3:32,54;
A169:Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
by A163,A165,A167,JGRAPH_2:11,JGRAPH_3:14;
A170: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2
= pz`2/sqrt(1+(pz`1/pz`2)^2) &
(|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1
= pz`1/sqrt(1+(pz`1/pz`2)^2) by EUCLID:56;
A171:(pz`2)^2 >=0 by SQUARE_1:72;
(pz`1/pz`2)^2 >=0 by SQUARE_1:72;
then 1+(pz`1/pz`2)^2>=1+0 by REAL_1:55;
then A172:1+(pz`1/pz`2)^2>0 by AXIOMS:22;
(|.p2.|)^2>=|.p2.| by A140,JGRAPH_2:2;
then A173: (|.p2.|)^2>=1 by A140,AXIOMS:22;
pz`2<>0 by A163,A165,A167;
then A174: (pz`2)^2<>0 by SQUARE_1:73;
A175:(|.p2.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by A168,A169,A170,JGRAPH_3:10
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2
+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A172,SQUARE_1:def 4
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2)
by A172,SQUARE_1:def 4
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)
by XCMPLX_1:63;
now
((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)
>=(1)*(1+(pz`1/pz`2)^2)
by A172,A173,A175,AXIOMS:25;
then ((pz`2)^2+(pz`1)^2)>=(1+(pz`1/pz`2)^2) by A172,XCMPLX_1:88;
then (pz`2)^2+(pz`1)^2>=1+(pz`1)^2/(pz`2)^2 by SQUARE_1:69;
then (pz`2)^2+(pz`1)^2-1>=1+(pz`1)^2/(pz`2)^2-1 by REAL_1:49;
then (pz`2)^2+(pz`1)^2-1>=(pz`1)^2/(pz`2)^2 by XCMPLX_1:26;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2>=(pz`1)^2/(pz`2)^2*(pz`2)^2
by A171,AXIOMS:25;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2>=(pz`1)^2 by A174,XCMPLX_1:88;
then ((pz`2)^2+((pz`1)^2-1))*(pz`2)^2>=(pz`1)^2 by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2+((pz`1)^2-1)*(pz`2)^2>=(pz`1)^2 by XCMPLX_1:8;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2
>=(pz`1)^2-(pz`1)^2 by REAL_1:49;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2>=0 by XCMPLX_1:
14
;
then (pz`2)^2*(pz`2)^2+((pz`2)^2*(pz`1)^2-(pz`2)^2*1)-(pz`1)^2>=0
by XCMPLX_1:40;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*(pz`1)^2-(pz`2)^2*1-(pz`1)^2>=0
by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2-(pz`2)^2*1+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2>=0
by XCMPLX_1:29;hence
(pz`2)^2*((pz`2)^2-1)+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2>=0
by XCMPLX_1:40;
end;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2*(pz`1)^2-(1)*(pz`1)^2)>=0
by XCMPLX_1:29;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2-1)*(pz`1)^2>=0
by XCMPLX_1:40;
then A176:((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)>=0 by XCMPLX_1:8;
A177: ((pz`2)^2+(pz`1)^2)<>0 by A165,COMPLEX1:2;
(pz`1)^2>=0 by SQUARE_1:72;
then ((pz`2)^2+(pz`1)^2)>=0+0 by A171,REAL_1:55;
then ((pz`2)^2-1)>=0 by A176,A177,SQUARE_1:25;
then (pz`2-1)*(pz`2+1)>=0 by SQUARE_1:59,67;
hence -1>=(gg.r)`1 or (gg.r)`1>=1 or
-1 >=(gg.r)`2 or (gg.r)`2>=1 by Th1;
end;
hence
(-1>=(ff.r)`1 or (ff.r)`1>=1 or -1 >=(ff.r)`2 or (ff.r)`2>=1)
& (-1>=(gg.r)`1 or (gg.r)`1>=1 or -1 >=(gg.r)`2 or (gg.r)`2>=1)
by A102;
end;
(-1 <=(ff.O)`2 & (ff.O)`2 <= 1) &
(-1 <=(ff.I)`2 & (ff.I)`2 <= 1) &
(-1 <=(gg.O)`1 & (gg.O)`1 <= 1) &
(-1 <=(gg.I)`1 & (gg.I)`1 <= 1)
proof
A178: (ff.O)=(Sq_Circ").(f.O) by A3,FUNCT_1:22;
consider p1 being Point of TOP-REAL 2 such that
A179: f.O=p1 &( |.p1.|=1 & p1`2>=p1`1 & p1`2<=-p1`1) by A1;
A180:p1<>0.REAL 2 &
(p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2<=-p1`1)
by A179,TOPRNS_1:24;
then A181:Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2/p1`1)
^2)]|
by JGRAPH_3:38;
reconsider px=ff.O as Point of TOP-REAL 2;
set q=px;
A182: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) &
px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2) by A178,A179,A181,EUCLID:56;
(p1`2/p1`1)^2 >=0 by SQUARE_1:72;
then 1+(p1`2/p1`1)^2>=1+0 by REAL_1:55;
then 1+(p1`2/p1`1)^2>0 by AXIOMS:22;
then A183:sqrt(1+(p1`2/p1`1)^2)>0 by SQUARE_1:93;
A184:now assume
A185: px`1=0 & px`2=0;
then A186:p1`1=0 by A182,A183,XCMPLX_1:6;
p1`2=0 by A182,A183,A185,XCMPLX_1:6;
hence contradiction by A180,A186,EUCLID:57,58;
end;
p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 &
p1`2*sqrt(1+(p1`2/p1`1)^2) <= (-p1`1)*sqrt(1+(p1`2/p1`1)^2)
by A179,A183,AXIOMS:25;
then p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2/p1`1
)^2)
or px`2>=px`1 & px`2<=-px`1 by A182,A183,AXIOMS:25,XCMPLX_1:175;
then A187:px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1
by A182,A183,AXIOMS:25,XCMPLX_1:175;
A188:p1=Sq_Circ.px by A178,A179,FUNCT_1:54,JGRAPH_3:32,54;
A189:Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|
by A184,A187,JGRAPH_2:11,JGRAPH_3:def 1;
A190: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1
= q`1/sqrt(1+(q`2/q`1)^2) &
(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2
= q`2/sqrt(1+(q`2/q`1)^2) by EUCLID:56;
(q`2/q`1)^2 >=0 by SQUARE_1:72;
then 1+(q`2/q`1)^2>=1+0 by REAL_1:55;
then A191:1+(q`2/q`1)^2>0 by AXIOMS:22;
px`1<>0 by A184,A187;
then A192: (px`1)^2<>0 by SQUARE_1:73;
now
(|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by A188,A189,A190,JGRAPH_3:10
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by SQUARE_1:69
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by A191,SQUARE_1:def 4
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A191,SQUARE_1:def 4
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:63;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)=(1)*(1+(q`2/q`1
)^2)
by A179,SQUARE_1:59;
then ((q`1)^2+(q`2)^2)=(1+(q`2/q`1)^2) by A191,XCMPLX_1:88;
then (px`1)^2+(px`2)^2=1+(px`2)^2/(px`1)^2 by SQUARE_1:69;
then (px`1)^2+(px`2)^2-1=(px`2)^2/(px`1)^2 by XCMPLX_1:26;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2=(px`2)^2 by A192,XCMPLX_1:88;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2=(px`2)^2 by XCMPLX_1:29;
then (px`1)^2*(px`1)^2+((px`2)^2-1)*(px`1)^2=(px`2)^2 by XCMPLX_1:8;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2=0 by XCMPLX_1:14
;
then (px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2=0
by XCMPLX_1:40;
then (px`1)^2*(px`1)^2+(px`1)^2*(px`2)^2-(px`1)^2*1-(px`2)^2=0
by XCMPLX_1:29;
then (px`1)^2*(px`1)^2-(px`1)^2*1+(px`1)^2*(px`2)^2-(1)*(px`2)^2=0
by XCMPLX_1:29;
then (px`1)^2*((px`1)^2-1)+(px`1)^2*(px`2)^2-(1)*(px`2)^2=0
by XCMPLX_1:40;
then (px`1)^2*((px`1)^2-1)+((px`1)^2*(px`2)^2-(1)*(px`2)^2)=0
by XCMPLX_1:29;
hence
((px`1)^2-1)*(px`1)^2+((px`1)^2-1)*(px`2)^2=0 by XCMPLX_1:40;
end;
then A193:((px`1)^2-1)*((px`1)^2+(px`2)^2)=0 by XCMPLX_1:8;
((px`1)^2+(px`2)^2)<>0 by A184,COMPLEX1:2;
then ((px`1)^2-1)=0 by A193,XCMPLX_1:6;
then (px`1-1)*(px`1+1)=0 by SQUARE_1:59,67;
then px`1-1=0 or px`1+1=0 by XCMPLX_1:6;
then px`1=0+1 or px`1+1=0 by XCMPLX_1:27;
then A194: px`1=1 or px`1=0-1 by XCMPLX_1:26;
A195: (ff.I)=(Sq_Circ").(f.I) by A3,FUNCT_1:22;
consider p2 being Point of TOP-REAL 2 such that
A196: f.I=p2 &(
|.p2.|=1 & p2`2<=p2`1 & p2`2>=-p2`1) by A1;
A197:p2<>0.REAL 2 &
(p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2<=-p2`1)
by A196,TOPRNS_1:24;
then A198:Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2/p2`1)
^2)]|
by JGRAPH_3:38;
reconsider py=ff.I as Point of TOP-REAL 2;
A199: py`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) &
py`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A195,A196,A198,EUCLID:56;
(p2`2/p2`1)^2 >=0 by SQUARE_1:72;
then 1+(p2`2/p2`1)^2>=1+0 by REAL_1:55;
then 1+(p2`2/p2`1)^2>0 by AXIOMS:22;
then A200:sqrt(1+(p2`2/p2`1)^2)>0 by SQUARE_1:93;
A201:now assume
A202: py`1=0 & py`2=0;
then A203:p2`1=0 by A199,A200,XCMPLX_1:6;
p2`2=0 by A199,A200,A202,XCMPLX_1:6;
hence contradiction by A197,A203,EUCLID:57,58;
end;
A204: now
p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/p2`1)^2)
or py`2>=py`1 & py`2<=-py`1 by A196,A200,AXIOMS:25;
hence
p2`2*sqrt(1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -py`1<=py
`2
or py`2>=py`1 & py`2<=-py`1 by A199,A200,AXIOMS:25,XCMPLX_1:175;
end;
A205:p2=Sq_Circ.py by A195,A196,FUNCT_1:54,JGRAPH_3:32,54;
A206:Sq_Circ.py=|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|
by A199,A201,A204,JGRAPH_2:11,JGRAPH_3:def 1;
A207: (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`1
= py`1/sqrt(1+(py`2/py`1)^2) &
(|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`2
= py`2/sqrt(1+(py`2/py`1)^2) by EUCLID:56;
(py`2/py`1)^2 >=0 by SQUARE_1:72;
then 1+(py`2/py`1)^2>=1+0 by REAL_1:55;
then A208:1+(py`2/py`1)^2>0 by AXIOMS:22;
py`1<>0 by A199,A201,A204;
then A209: (py`1)^2<>0 by SQUARE_1:73;
now
(|.p2.|)^2= (py`1/sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
by A205,A206,A207,JGRAPH_3:10
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
by SQUARE_1:69
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2
+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2
by SQUARE_1:69
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2
by A208,SQUARE_1:def 4
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(1+(py`2/py`1)^2)
by A208,SQUARE_1:def 4
.= ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2)
by XCMPLX_1:63;
then ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2)*(1+(py`2/py`1)^2)
=(1)*(1+(py`2/py`1)^2) by A196,SQUARE_1:59;
then ((py`1)^2+(py`2)^2)=(1+(py`2/py`1)^2)
by A208,XCMPLX_1:88;
then (py`1)^2+(py`2)^2=1+(py`2)^2/(py`1)^2 by SQUARE_1:69;
then (py`1)^2+(py`2)^2-1=(py`2)^2/(py`1)^2 by XCMPLX_1:26;
then ((py`1)^2+(py`2)^2-1)*(py`1)^2=(py`2)^2
by A209,XCMPLX_1:88;
then ((py`1)^2+((py`2)^2-1))*(py`1)^2=(py`2)^2 by XCMPLX_1:29;
then (py`1)^2*(py`1)^2+((py`2)^2-1)*(py`1)^2=(py`2)^2 by XCMPLX_1:8;
then (py`1)^2*(py`1)^2+(py`1)^2*((py`2)^2-1)-(py`2)^2=0 by XCMPLX_1:14
;
then (py`1)^2*(py`1)^2+((py`1)^2*(py`2)^2-(py`1)^2*1)-(py`2)^2=0
by XCMPLX_1:40;
then (py`1)^2*(py`1)^2+(py`1)^2*(py`2)^2-(py`1)^2*1-(py`2)^2=0
by XCMPLX_1:29;
then (py`1)^2*(py`1)^2-(py`1)^2*1+(py`1)^2*(py`2)^2-(1)*(py`2)^2=0
by XCMPLX_1:29;
then (py`1)^2*((py`1)^2-1)+(py`1)^2*(py`2)^2-(1)*(py`2)^2=0
by XCMPLX_1:40;
then (py`1)^2*((py`1)^2-1)+((py`1)^2*(py`2)^2-(1)*(py`2)^2)=0
by XCMPLX_1:29;
hence ((py`1)^2-1)*(py`1)^2+((py`1)^2-1)*(py`2)^2=0 by XCMPLX_1:40;
end;
then A210:((py`1)^2-1)*((py`1)^2+(py`2)^2)=0 by XCMPLX_1:8;
((py`1)^2+(py`2)^2)<>0 by A201,COMPLEX1:2;
then ((py`1)^2-1)=0 by A210,XCMPLX_1:6;
then (py`1-1)*(py`1+1)=0 by SQUARE_1:59,67;
then py`1-1=0 or py`1+1=0 by XCMPLX_1:6;
then py`1=0+1 or py`1+1=0 by XCMPLX_1:27;
then A211: py`1=1 or py`1=0-1 by XCMPLX_1:26;
A212: gg.O=(Sq_Circ").(g.O) by A4,FUNCT_1:22;
consider p3 being Point of TOP-REAL 2 such that
A213: g.O=p3 &( |.p3.|=1 & p3`2<=p3`1 & p3`2<=-p3`1) by A1;
A214: -p3`2>=--p3`1 by A213,REAL_1:50;
then A215:p3<>0.REAL 2 &
(p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2)
by A213,TOPRNS_1:24;
then A216:Sq_Circ".p3=|[p3`1*sqrt(1+(p3`1/p3`2)^2),p3`2*sqrt(1+(p3`1/p3`2)^2
)]|
by JGRAPH_3:40;
reconsider pz=gg.O as Point of TOP-REAL 2;
A217: pz`2 = p3`2*sqrt(1+(p3`1/p3`2)^2) &
pz`1 = p3`1*sqrt(1+(p3`1/p3`2)^2) by A212,A213,A216,EUCLID:56;
(p3`1/p3`2)^2 >=0 by SQUARE_1:72;
then 1+(p3`1/p3`2)^2>=1+0 by REAL_1:55;
then 1+(p3`1/p3`2)^2>0 by AXIOMS:22;
then A218:sqrt(1+(p3`1/p3`2)^2)>0 by SQUARE_1:93;
A219:now assume
A220: pz`2=0 & pz`1=0;
then A221:p3`2=0 by A217,A218,XCMPLX_1:6;
p3`1=0 by A217,A218,A220,XCMPLX_1:6;
hence contradiction by A215,A221,EUCLID:57,58;
end;
A222: now
p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 &
p3`1*sqrt(1+(p3`1/p3`2)^2) <= (-p3`2)*sqrt(1+(p3`1/p3`2)^2)
by A213,A214,A218,AXIOMS:25;
then p3`1<=p3`2 & (-p3`2)*sqrt(1+(p3`1/p3`2)^2) <= p3`1*sqrt(1+(p3`1/p3`2
)^2)
or pz`1>=pz`2 & pz`1<=-pz`2 by A217,A218,AXIOMS:25,XCMPLX_1:175;
hence
p3`1*sqrt(1+(p3`1/p3`2)^2) <= p3`2*sqrt(1+(p3`1/p3`2)^2) & -pz`2<=pz`1
or pz`1>=pz`2 & pz`1<=-pz`2 by A217,A218,AXIOMS:25,XCMPLX_1:175;
end;
A223:p3=Sq_Circ.pz by A212,A213,FUNCT_1:54,JGRAPH_3:32,54;
A224:Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
by A217,A219,A222,JGRAPH_2:11,JGRAPH_3:14;
A225: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2
= pz`2/sqrt(1+(pz`1/pz`2)^2) &
(|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1
= pz`1/sqrt(1+(pz`1/pz`2)^2) by EUCLID:56;
(pz`1/pz`2)^2 >=0 by SQUARE_1:72;
then 1+(pz`1/pz`2)^2>=1+0 by REAL_1:55;
then A226:1+(pz`1/pz`2)^2>0 by AXIOMS:22;
pz`2<>0 by A217,A219,A222;
then A227: (pz`2)^2<>0 by SQUARE_1:73;
A228:(|.p3.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by A223,A224,A225,JGRAPH_3:10
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2
+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by SQUARE_1:69
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A226,SQUARE_1:def 4
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2)
by A226,SQUARE_1:def 4
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)
by XCMPLX_1:63;
now
((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)=(1)*(1+(pz`1/pz`2)^2
)
by A213,A228,SQUARE_1:59;
then ((pz`2)^2+(pz`1)^2)=(1+(pz`1/pz`2)^2)
by A226,XCMPLX_1:88;
then (pz`2)^2+(pz`1)^2=1+(pz`1)^2/(pz`2)^2 by SQUARE_1:69;
then (pz`2)^2+(pz`1)^2-1=(pz`1)^2/(pz`2)^2 by XCMPLX_1:26;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2=(pz`1)^2
by A227,XCMPLX_1:88;
then ((pz`2)^2+((pz`1)^2-1))*(pz`2)^2=(pz`1)^2 by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2+((pz`1)^2-1)*(pz`2)^2=(pz`1)^2 by XCMPLX_1:8;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2=0 by XCMPLX_1:14
;
then (pz`2)^2*(pz`2)^2+((pz`2)^2*(pz`1)^2-(pz`2)^2*1)-(pz`1)^2=0
by XCMPLX_1:40;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*(pz`1)^2-(pz`2)^2*1-(pz`1)^2=0
by XCMPLX_1:29;
then (pz`2)^2*(pz`2)^2-(pz`2)^2*1+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2=0
by XCMPLX_1:29;hence
(pz`2)^2*((pz`2)^2-1)+(pz`2)^2*(pz`1)^2-(1)*(pz`1)^2=0
by XCMPLX_1:40;
end;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2*(pz`1)^2-(1)*(pz`1)^2)=0
by XCMPLX_1:29;
then (pz`2)^2*((pz`2)^2-1)+((pz`2)^2-1)*(pz`1)^2=0
by XCMPLX_1:40;
then A229:((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)=0 by XCMPLX_1:8;
((pz`2)^2+(pz`1)^2)<>0 by A219,COMPLEX1:2;
then ((pz`2)^2-1)=0 by A229,XCMPLX_1:6;
then (pz`2-1)*(pz`2+1)=0 by SQUARE_1:59,67;
then pz`2-1=0 or pz`2+1=0 by XCMPLX_1:6;
then pz`2=0+1 or pz`2+1=0 by XCMPLX_1:27;
then A230: pz`2=1 or pz`2=0-1 by XCMPLX_1:26;
A231: (gg.I)=(Sq_Circ").(g.I) by A4,FUNCT_1:22;
consider p4 being Point of TOP-REAL 2 such that
A232: g.I=p4 &(
|.p4.|=1 & p4`2>=p4`1 & p4`2>=-p4`1) by A1;
A233: -p4`2<=--p4`1 by A232,REAL_1:50;
then A234:p4<>0.REAL 2 &
(p4`1<=p4`2 & -p4`2<=p4`1 or p4`1>=p4`2 & p4`1<=-p4`2)
by A232,TOPRNS_1:24;
then A235:Sq_Circ".p4=|[p4`1*sqrt(1+(p4`1/p4`2)^2),p4`2*sqrt(1+(p4`1/p4`2)
^2)]|
by JGRAPH_3:40;
reconsider pu=gg.I as Point of TOP-REAL 2;
A236: pu`2 = p4`2*sqrt(1+(p4`1/p4`2)^2) &
pu`1 = p4`1*sqrt(1+(p4`1/p4`2)^2) by A231,A232,A235,EUCLID:56;
(p4`1/p4`2)^2 >=0 by SQUARE_1:72;
then 1+(p4`1/p4`2)^2>=1+0 by REAL_1:55;
then 1+(p4`1/p4`2)^2>0 by AXIOMS:22;
then A237:sqrt(1+(p4`1/p4`2)^2)>0 by SQUARE_1:93;
A238:now assume
A239: pu`2=0 & pu`1=0;
then A240:p4`2=0 by A236,A237,XCMPLX_1:6;
p4`1=0 by A236,A237,A239,XCMPLX_1:6;
hence contradiction by A234,A240,EUCLID:57,58;
end;
A241: now
p4`1<=p4`2 & (-p4`2)*sqrt(1+(p4`1/p4`2)^2) <= p4`1*sqrt(1+(p4`1/p4`2)^2)
or pu`1>=pu`2 & pu`1<=-pu`2 by A232,A233,A237,AXIOMS:25;hence
p4`1*sqrt(1+(p4`1/p4`2)^2) <= p4`2*sqrt(1+(p4`1/p4`2)^2) & -pu`2<=pu`1
or pu`1>=pu`2 & pu`1<=-pu`2 by A236,A237,AXIOMS:25,XCMPLX_1:175;
end;
A242:p4=Sq_Circ.pu by A231,A232,FUNCT_1:54,JGRAPH_3:32,54;
A243:Sq_Circ.pu=|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|
by A236,A238,A241,JGRAPH_2:11,JGRAPH_3:14;
A244: (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`2
= pu`2/sqrt(1+(pu`1/pu`2)^2) &
(|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`1
= pu`1/sqrt(1+(pu`1/pu`2)^2) by EUCLID:56;
(pu`1/pu`2)^2 >=0 by SQUARE_1:72;
then 1+(pu`1/pu`2)^2>=1+0 by REAL_1:55;
then A245:1+(pu`1/pu`2)^2>0 by AXIOMS:22;
pu`2<>0 by A236,A238,A241;
then A246: (pu`2)^2<>0 by SQUARE_1:73;
now
(|.p4.|)^2= (pu`2/sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
by A242,A243,A244,JGRAPH_3:10
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
by SQUARE_1:69
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2
+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2
by SQUARE_1:69
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2
by A245,SQUARE_1:def 4
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(1+(pu`1/pu`2)^2)
by A245,SQUARE_1:def 4
.= ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2) by XCMPLX_1:63;
then ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2)*(1+(pu`1/pu`2)^2)=(1)*(1+(pu`1/pu
`2)^2)
by A232,SQUARE_1:59;
then ((pu`2)^2+(pu`1)^2)=(1+(pu`1/pu`2)^2) by A245,XCMPLX_1:88;
then (pu`2)^2+(pu`1)^2=1+(pu`1)^2/(pu`2)^2 by SQUARE_1:69;
then (pu`2)^2+(pu`1)^2-1=(pu`1)^2/(pu`2)^2 by XCMPLX_1:26;
then ((pu`2)^2+(pu`1)^2-1)*(pu`2)^2=(pu`1)^2 by A246,XCMPLX_1:88;
then ((pu`2)^2+((pu`1)^2-1))*(pu`2)^2=(pu`1)^2 by XCMPLX_1:29;
then (pu`2)^2*(pu`2)^2+((pu`1)^2-1)*(pu`2)^2=(pu`1)^2 by XCMPLX_1:8;
then (pu`2)^2*(pu`2)^2+(pu`2)^2*((pu`1)^2-1)-(pu`1)^2=0 by XCMPLX_1:14
;
then (pu`2)^2*(pu`2)^2+((pu`2)^2*(pu`1)^2-(pu`2)^2*1)-(pu`1)^2=0
by XCMPLX_1:40;
then (pu`2)^2*(pu`2)^2+(pu`2)^2*(pu`1)^2-(pu`2)^2*1-(pu`1)^2=0
by XCMPLX_1:29;
then (pu`2)^2*(pu`2)^2-(pu`2)^2*1+(pu`2)^2*(pu`1)^2-(1)*(pu`1)^2=0
by XCMPLX_1:29;
then (pu`2)^2*((pu`2)^2-1)+(pu`2)^2*(pu`1)^2-(1)*(pu`1)^2=0
by XCMPLX_1:40;
then (pu`2)^2*((pu`2)^2-1)+((pu`2)^2*(pu`1)^2-(1)*(pu`1)^2)=0
by XCMPLX_1:29;
hence ((pu`2)^2-1)*(pu`2)^2+((pu`2)^2-1)*(pu`1)^2=0 by XCMPLX_1:40;
end;
then A247:((pu`2)^2-1)*((pu`2)^2+(pu`1)^2)=0 by XCMPLX_1:8;
((pu`2)^2+(pu`1)^2)<>0 by A238,COMPLEX1:2;
then (pu`2)^2-1=0 by A247,XCMPLX_1:6;
then (pu`2-1)*(pu`2+1)=0 by SQUARE_1:59,67;
then pu`2-1=0 or pu`2+1=0 by XCMPLX_1:6;
then pu`2=0+1 or pu`2+1=0 by XCMPLX_1:27;
then A248: pu`2=1 or pu`2=0-1 by XCMPLX_1:26;
thus -1 <=(ff.O)`2 & (ff.O)`2 <= 1 by A187,A194,AXIOMS:22;
thus -1 <=(ff.I)`2 & (ff.I)`2 <= 1 by A199,A204,A211,AXIOMS:22;
thus -1 <=(gg.O)`1 & (gg.O)`1 <= 1 by A217,A222,A230,AXIOMS:22;
thus -1 <=(gg.I)`1 & (gg.I)`1 <= 1 by A236,A241,A248,AXIOMS:22;
end;
then rng ff meets rng gg by A1,A7,A9,A10,A11,A12,A99,Th14;
then consider y being set such that
A249: y in rng ff & y in rng gg by XBOOLE_0:3;
consider x1 being set such that
A250: x1 in dom ff & y=ff.x1 by A249,FUNCT_1:def 5;
consider x2 being set such that
A251: x2 in dom gg & y=gg.x2 by A249,FUNCT_1:def 5;
A252:ff.x1=Sq_Circ".(f.x1) by A250,FUNCT_1:22;
A253:dom (Sq_Circ")=the carrier of TOP-REAL 2
by FUNCT_2:def 1,JGRAPH_3:39;
x1 in dom f by A250,FUNCT_1:21;
then A254:f.x1 in rng f by FUNCT_1:def 5;
x2 in dom g by A251,FUNCT_1:21;
then A255:g.x2 in rng g by FUNCT_1:def 5;
A256: Sq_Circ" is one-to-one by FUNCT_1:62,JGRAPH_3:32;
gg.x2=Sq_Circ".(g.x2) by A251,FUNCT_1:22;
then f.x1=g.x2 by A250,A251,A252,A253,A254,A255,A256,FUNCT_1:def 8;
hence thesis by A254,A255,XBOOLE_0:3;
end;
theorem Th18: for f,g being map of I[01],TOP-REAL 2,
C0,KXP,KXN,KYP,KYN being Subset of TOP-REAL 2,
O,I being Point of I[01] st O=0 & I=1 &
f is continuous one-to-one &
g is continuous one-to-one &
C0={p: |.p.|>=1}&
KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} &
KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} &
KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} &
KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} &
f.O in KXN & f.I in KXP &
g.O in KYP & g.I in KYN &
rng f c= C0 & rng g c= C0
holds rng f meets rng g
proof let f,g be map of I[01],TOP-REAL 2,
C0,KXP,KXN,KYP,KYN be Subset of TOP-REAL 2,
O,I be Point of I[01];
assume A1: O=0 & I=1 &
f is continuous one-to-one & g is continuous one-to-one &
C0={p: |.p.|>=1}&
KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} &
KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} &
KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} &
KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} &
f.O in KXN & f.I in KXP &
g.O in KYP & g.I in KYN &
rng f c= C0 & rng g c= C0;
then consider g2 being map of I[01],TOP-REAL 2 such that
A2: g2.0=g.1 & g2.1=g.0 &
rng g2=rng g & g2 is continuous one-to-one by Th15;
thus rng f meets rng g by A1,A2,Th17;
end;
theorem Th19:
for f,g being map of I[01],TOP-REAL 2,
C0 being Subset of TOP-REAL 2
st C0={q: |.q.|>=1} &
f is continuous one-to-one &
g is continuous one-to-one &
f.0=|[-1,0]| & f.1=|[1,0]| & g.1=|[0,1]| & g.0=|[0,-1]|
& rng f c= C0 & rng g c= C0
holds rng f meets rng g
proof let f,g be map of I[01],TOP-REAL 2,
C0 be Subset of TOP-REAL 2;
assume A1: C0={q: |.q.|>=1} &
f is continuous one-to-one & g is continuous one-to-one &
f.0=|[-1,0]| & f.1=|[1,0]| & g.1=|[0,1]| & g.0=|[0,-1]|
& rng f c= C0 & rng g c= C0;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1] } is Subset
of TOP-REAL 2 from TopSubset;
then reconsider KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
{q2 where q2 is Point of TOP-REAL 2: P[q2]}
is Subset of TOP-REAL 2 from TopSubset;
then reconsider KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]}
is Subset of TOP-REAL 2 from TopSubset;
then reconsider KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]}
is Subset of TOP-REAL 2 from TopSubset;
then reconsider KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} as Subset of TOP-REAL 2;
reconsider O=0 as Point of I[01] by BORSUK_1:83,RCOMP_1:15;
reconsider I=1 as Point of I[01] by BORSUK_1:83,RCOMP_1:15;
A2: (|[-1,0]|)`1=-1 & (|[-1,0]|)`2=0 by EUCLID:56;
then A3: |. (|[-1,0]|).|=sqrt((-1)^2+0^2) by JGRAPH_3:10
.=1 by SQUARE_1:59,60,61,83;
(|[-1,0]|)`2 <=-((|[-1,0]|)`1) by A2;
then A4: f.O in KXN by A1,A2,A3;
A5: (|[1,0]|)`1=1 & (|[1,0]|)`2=0 by EUCLID:56;
then |.(|[1,0]|).|=sqrt(1^2+0) by JGRAPH_3:10,SQUARE_1:60
.=1 by SQUARE_1:59,83;
then A6: f.I in KXP by A1,A5;
A7: (|[0,-1]|)`2=-1 & (|[0,-1]|)`1=0 by EUCLID:56;
then A8: |.(|[0,-1]|).|=sqrt(0^2+(-1)^2) by JGRAPH_3:10
.=1 by SQUARE_1:59,60,61,83;
(|[0,-1]|)`2 <=-((|[0,-1]|)`1) by A7;
then A9: g.O in KYN by A1,A7,A8;
A10: (|[0,1]|)`2=1 & (|[0,1]|)`1=0 by EUCLID:56;
then A11: |. (|[0,1]|).|=sqrt(0+1^2) by JGRAPH_3:10,SQUARE_1:60
.=1 by SQUARE_1:59,83;
(|[0,1]|)`2 >=-((|[0,1]|)`1) by A10;
then g.I in KYP by A1,A10,A11;
hence rng f meets rng g by A1,A4,A6,A9,Th17;
end;
theorem for p1,p2,p3,p4 being Point of TOP-REAL 2,
C0 being Subset of TOP-REAL 2
st C0={p: |.p.|>=1}
& |.p1.|=1 & |.p2.|=1 & |.p3.|=1 & |.p4.|=1 &
(ex h being map of TOP-REAL 2,TOP-REAL 2 st h is_homeomorphism
& h.:C0 c= C0 &
h.p1=|[-1,0]| & h.p2=|[0,1]| & h.p3=|[1,0]| & h.p4=|[0,-1]|)
holds
(for f,g being map of I[01],TOP-REAL 2 st
f is continuous one-to-one &
g is continuous one-to-one &
f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 &
rng f c= C0 & rng g c= C0
holds rng f meets rng g)
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
C0 be Subset of TOP-REAL 2;
assume A1: C0={p: |.p.|>=1}
& |.p1.|=1 & |.p2.|=1 & |.p3.|=1 & |.p4.|=1 &
(ex h being map of TOP-REAL 2,TOP-REAL 2 st h is_homeomorphism
& h.:C0 c= C0 &
h.p1=(|[-1,0]|) & h.p2=(|[0,1]|) & h.p3=(|[1,0]|) & h.p4=(|[0,-1]|));
then consider h being map of TOP-REAL 2,TOP-REAL 2 such that
A2: h is_homeomorphism
& h.:C0 c= C0 &
h.p1=(|[-1,0]|) & h.p2=(|[0,1]|) & h.p3=(|[1,0]|) & h.p4=(|[0,-1]|);
let f,g be map of I[01],TOP-REAL 2;
assume A3: f is continuous one-to-one &
g is continuous one-to-one &
f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 &
rng f c= C0 & rng g c= C0;
reconsider f2=h*f as map of I[01],TOP-REAL 2;
reconsider g2=h*g as map of I[01],TOP-REAL 2;
A4: h is continuous by A2,TOPS_2:def 5;
then A5: f2 is continuous by A3,TOPS_2:58;
A6: g2 is continuous by A3,A4,TOPS_2:58;
A7: h is one-to-one by A2,TOPS_2:def 5;
then A8: f2 is one-to-one by A3,FUNCT_1:46;
A9: g2 is one-to-one by A3,A7,FUNCT_1:46;
A10: 0 in dom f2 &1 in dom f2 by Lm1,BORSUK_1:83,FUNCT_2:def 1;
then A11: f2.0=|[-1,0]| by A2,A3,FUNCT_1:22;
A12: f2.1=|[1,0]| by A2,A3,A10,FUNCT_1:22;
A13: 0 in dom g2 &1 in dom g2 by Lm1,BORSUK_1:83,FUNCT_2:def 1;
then A14: g2.0=|[0,-1]| by A2,A3,FUNCT_1:22;
A15: g2.1=|[0,1]| by A2,A3,A13,FUNCT_1:22;
A16: rng f2 c= C0
proof let y be set;assume y in rng f2;
then consider x being set such that
A17: x in dom f2 & y=f2.x by FUNCT_1:def 5;
A18: y=h.(f.x) by A17,FUNCT_1:22;
x in dom f by A17,FUNCT_1:21;
then A19: f.x in rng f by FUNCT_1:def 5;
dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then y in h.:C0 by A3,A18,A19,FUNCT_1:def 12;
hence y in C0 by A2;
end;
rng g2 c= C0
proof let y be set;assume y in rng g2;
then consider x being set such that
A20: x in dom g2 & y=g2.x by FUNCT_1:def 5;
A21: y=h.(g.x) by A20,FUNCT_1:22;
x in dom g by A20,FUNCT_1:21;
then A22: g.x in rng g by FUNCT_1:def 5;
dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then y in h.:C0 by A3,A21,A22,FUNCT_1:def 12;
hence y in C0 by A2;
end;
then rng f2 meets rng g2 by A1,A5,A6,A8,A9,A11,A12,A14,A15,A16,Th19;
then consider q5 being set such that
A23: q5 in rng f2 & q5 in rng g2 by XBOOLE_0:3;
consider x being set such that
A24: x in dom f2 & q5=f2.x by A23,FUNCT_1:def 5;
A25: q5=h.(f.x) by A24,FUNCT_1:22;
consider u being set such that
A26: u in dom g2 & q5=g2.u by A23,FUNCT_1:def 5;
A27: q5=h.(g.u) by A26,FUNCT_1:22;
A28: h is one-to-one by A2,TOPS_2:def 5;
A29: f.x in dom h by A24,FUNCT_1:21;
g.u in dom h by A26,FUNCT_1:21;
then A30: f.x=g.u by A25,A27,A28,A29,FUNCT_1:def 8;
A31: x in dom f by A24,FUNCT_1:21;
A32: u in dom g by A26,FUNCT_1:21;
A33: f.x in rng f by A31,FUNCT_1:def 5;
g.u in rng g by A32,FUNCT_1:def 5;
hence thesis by A30,A33,XBOOLE_0:3;
end;
begin :: Properties of Fan Morphisms
theorem Th21:
for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2>0
holds (for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>0)
proof let cn be Real,q be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q`2>0;
now per cases;
case q`1/|.q.|>=cn;
hence (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphN).q holds p`2>0) by A1,JGRAPH_4:82;
case q`1/|.q.|<cn;
hence (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphN).q holds p`2>0) by A1,JGRAPH_4:83;
end;
hence thesis;
end;
theorem
for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2>=0
holds (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphN).q holds p`2>=0)
proof let cn be Real,q be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q`2>=0;
now per cases by A1;
case q`2>0;
hence (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphN).q holds p`2>=0) by A1,Th21;
case q`2=0;
hence (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphN).q holds p`2>=0) by JGRAPH_4:56;
end;
hence thesis;
end;
theorem Th23:
for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2>=0
& q`1/|.q.|<cn & |.q.|<>0 holds (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphN).q holds p`2>=0 & p`1<0)
proof let cn be Real,q be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q`2>=0
& q`1/|.q.|<cn & |.q.|<>0;
let p be Point of TOP-REAL 2;
assume A2: p=(cn-FanMorphN).q;
now per cases;
case A3: q`2=0;
then A4: q=p by A2,JGRAPH_4:56;
|.q.|^2=(q`1)^2+0 by A3,JGRAPH_3:10,SQUARE_1:60 .=(q`1)^2;
then |.q.|=q`1 or |.q.|=-(q`1) by JGRAPH_3:1;
then (-(q`1))/|.q.|=1 by A1,XCMPLX_1:60;
then -(q`1/|.q.|)=1 by XCMPLX_1:188;
then A5: q`1=(-1)*|.q.| by A1,XCMPLX_1:88;
|.q.|>=0 by TOPRNS_1:26;
hence p`2>=0 & p`1<0 by A1,A4,A5,SQUARE_1:24;
case q`2<>0;
hence p`2>=0 & p`1<0 by A1,A2,JGRAPH_4:83;
end;
hence p`2>=0 & p`1<0;
end;
theorem Th24:
for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & cn<1 & q1`2>=0
& q2`2>=0 & |.q1.|<>0 & |.q2.|<>0 & q1`1/|.q1.|<q2`1/|.q2.|
holds (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2 holds p1`1/|.p1.|<p2`1/|.p2.|)
proof let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q1`2>=0 & q2`2>=0 & |.q1.|<>0 & |.q2.|<>0
& q1`1/|.q1.|<q2`1/|.q2.|;
now per cases by A1;
case A2: q1`2>0;
now per cases by A1;
case q2`2>0;
hence (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2 holds
p1`1/|.p1.|<p2`1/|.p2.|) by A1,A2,JGRAPH_4:86;
case A3: q2`2=0;
then |.q2.|^2=(q2`1)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(q2`1)^2;
then A4: |.q2.|=q2`1 or |.q2.|=-(q2`1) by JGRAPH_3:1;
now assume |.q2.|=-(q2`1);
then 1=(-(q2`1))/|.q2.| by A1,XCMPLX_1:60;
then A5: q1`1/|.q1.|< -1 by A1,XCMPLX_1:191;
A6: |.q1.|>=0 by TOPRNS_1:26;
(|.q1.|)^2=(q1`1)^2+(q1`2)^2 by JGRAPH_3:10;
then (|.q1.|)^2-(q1`1)^2=(q1`2)^2 by XCMPLX_1:26;
then (|.q1.|)^2-(q1`1)^2>=0 by SQUARE_1:72;
then (|.q1.|)^2-(q1`1)^2+(q1`1)^2>=0+(q1`1)^2 by REAL_1:55;
then (|.q1.|)^2>=(q1`1)^2 by XCMPLX_1:27;
then -|.q1.|<=q1`1 & q1`1<=|.q1.| by A6,JGRAPH_2:5;
then (-|.q1.|)/|.q1.|<=q1`1/|.q1.| by A1,A6,REAL_1:73;
hence contradiction by A1,A5,XCMPLX_1:198;
end;
then A7: q2`1/|.q2.|=1 by A1,A4,XCMPLX_1:60;
thus for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2 holds
p1`1/|.p1.|<p2`1/|.p2.|
proof let p1,p2 be Point of TOP-REAL 2;
assume A8: p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2;
then A9: p2=q2 by A3,JGRAPH_4:56;
A10: |.p1.|=|.q1.| by A8,JGRAPH_4:73;
A11: |.p1.|>=0 by TOPRNS_1:26;
(|.p1.|)^2=(p1`1)^2+(p1`2)^2 by JGRAPH_3:10;
then A12: (|.p1.|)^2-(p1`1)^2=(p1`2)^2 by XCMPLX_1:26;
then (|.p1.|)^2-(p1`1)^2>=0 by SQUARE_1:72;
then (|.p1.|)^2-(p1`1)^2+(p1`1)^2>=0+(p1`1)^2 by REAL_1:55;
then (|.p1.|)^2>=(p1`1)^2 by XCMPLX_1:27;
then -|.p1.|<=p1`1 & p1`1<=|.p1.| by A11,JGRAPH_2:5;
then (|.p1.|)/|.p1.|>=p1`1/|.p1.| by A1,A10,A11,REAL_1:73;
then A13: 1>= p1`1/|.p1.| by A1,A10,XCMPLX_1:60;
A14: p1`2>0 by A1,A2,A8,Th21;
now assume 1= p1`1/|.p1.|; then (1)*|.p1.|=p1`1 by A1,A10,XCMPLX_1:
88
;
then (|.p1.|)^2-(p1`1)^2=0 by XCMPLX_1:14;
hence contradiction by A12,A14,SQUARE_1:73;
end;
hence p1`1/|.p1.|<p2`1/|.p2.| by A7,A9,A13,REAL_1:def 5;
end;
end;
hence (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2 holds
p1`1/|.p1.|<p2`1/|.p2.|);
case A15: q1`2=0;
then |.q1.|^2=(q1`1)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(q1`1)^2;
then A16: |.q1.|=q1`1 or |.q1.|=-(q1`1) by JGRAPH_3:1;
now assume |.q1.|=(q1`1);
then A17: q2`1/|.q2.|> 1 by A1,XCMPLX_1:60;
A18: |.q2.|>=0 by TOPRNS_1:26;
(|.q2.|)^2=(q2`1)^2+(q2`2)^2 by JGRAPH_3:10;
then (|.q2.|)^2-(q2`1)^2=(q2`2)^2 by XCMPLX_1:26;
then (|.q2.|)^2-(q2`1)^2>=0 by SQUARE_1:72;
then (|.q2.|)^2-(q2`1)^2+(q2`1)^2>=0+(q2`1)^2 by REAL_1:55;
then (|.q2.|)^2>=(q2`1)^2 by XCMPLX_1:27;
then -|.q2.|<=q2`1 & q2`1<=|.q2.| by A18,JGRAPH_2:5;
then (|.q2.|)/|.q2.|>=q2`1/|.q2.| by A1,A18,REAL_1:73;
hence contradiction by A1,A17,XCMPLX_1:60;
end;
then (-(q1`1))/|.q1.|=1 by A1,A16,XCMPLX_1:60;
then A19: -(q1`1/|.q1.|)=1 by XCMPLX_1:188;
thus (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2
holds p1`1/|.p1.|<p2`1/|.p2.|)
proof let p1,p2 be Point of TOP-REAL 2;
assume A20: p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2;
then A21: p1=q1 by A15,JGRAPH_4:56;
A22: |.p2.|=|.q2.| by A20,JGRAPH_4:73;
A23: |.p2.|>=0 by TOPRNS_1:26;
(|.p2.|)^2=(p2`1)^2+(p2`2)^2 by JGRAPH_3:10;
then A24: (|.p2.|)^2-(p2`1)^2=(p2`2)^2 by XCMPLX_1:26;
then (|.p2.|)^2-(p2`1)^2>=0 by SQUARE_1:72;
then (|.p2.|)^2-(p2`1)^2+(p2`1)^2>=0+(p2`1)^2 by REAL_1:55;
then (|.p2.|)^2>=(p2`1)^2 by XCMPLX_1:27;
then -|.p2.|<=p2`1 & p2`1<=|.p2.| by A23,JGRAPH_2:5;
then (-|.p2.|)/|.p2.|<=p2`1/|.p2.| by A1,A22,A23,REAL_1:73;
then A25: -1 <= p2`1/|.p2.| by A1,A22,XCMPLX_1:198;
now per cases;
case q2`2=0;
then p2=q2 by A20,JGRAPH_4:56;
hence p2`1/|.p2.|>-1 by A1,A19;
case q2`2<>0;
then A26: p2`2>0 by A1,A20,Th21;
now assume -1= p2`1/|.p2.|; then (-1)*|.p2.|=p2`1 by A1,A22,XCMPLX_1
:88;
then (-|.p2.|)^2=(p2`1)^2 by XCMPLX_1:180;
then (|.p2.|)^2=(p2`1)^2 by SQUARE_1:61;
then (|.p2.|)^2-(p2`1)^2=0 by XCMPLX_1:14;
hence contradiction by A24,A26,SQUARE_1:73;
end;
hence p2`1/|.p2.|>-1 by A25,REAL_1:def 5;
end;
hence p1`1/|.p1.|<p2`1/|.p2.| by A19,A21;
end;
end;
hence thesis;
end;
theorem Th25:
for sn being Real,q being Point of TOP-REAL 2 st -1<sn & sn<1 & q`1>0
holds (for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds p`1>0)
proof let sn be Real,q be Point of TOP-REAL 2;
assume A1: -1<sn & sn<1 & q`1>0;
now per cases;
case q`2/|.q.|>=sn;
hence (for p being Point of TOP-REAL 2 st
p=(sn-FanMorphE).q holds p`1>0) by A1,JGRAPH_4:113;
case q`2/|.q.|<sn;
hence (for p being Point of TOP-REAL 2 st
p=(sn-FanMorphE).q holds p`1>0) by A1,JGRAPH_4:114;
end;
hence thesis;
end;
theorem
for sn being Real,q being Point of TOP-REAL 2 st -1<sn & sn<1 & q`1>=0
& q`2/|.q.|<sn & |.q.|<>0 holds (for p being Point of TOP-REAL 2 st
p=(sn-FanMorphE).q holds p`1>=0 & p`2<0)
proof let sn be Real,q be Point of TOP-REAL 2;
assume A1: -1<sn & sn<1 & q`1>=0 & q`2/|.q.|<sn & |.q.|<>0;
let p be Point of TOP-REAL 2;
assume A2: p=(sn-FanMorphE).q;
now per cases;
case A3: q`1=0;
then A4: q=p by A2,JGRAPH_4:89;
|.q.|^2=(q`2)^2+0 by A3,JGRAPH_3:10,SQUARE_1:60 .=(q`2)^2;
then |.q.|=q`2 or |.q.|=-(q`2) by JGRAPH_3:1;
then (-(q`2))/|.q.|=1 by A1,XCMPLX_1:60;
then -(q`2/|.q.|)=1 by XCMPLX_1:188;
then A5: q`2=(-1)*|.q.| by A1,XCMPLX_1:88;
|.q.|>=0 by TOPRNS_1:26;
hence p`1>=0 & p`2<0 by A1,A4,A5,SQUARE_1:24;
case q`1<>0;
hence p`1>=0 & p`2<0 by A1,A2,JGRAPH_4:114;
end;
hence p`1>=0 & p`2<0;
end;
theorem Th27:
for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1<sn & sn<1 & q1`1>=0
& q2`1>=0 & |.q1.|<>0 & |.q2.|<>0 & q1`2/|.q1.|<q2`2/|.q2.|
holds (for p1,p2 being Point of TOP-REAL 2 st
p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2 holds p1`2/|.p1.|<p2`2/|.p2.|)
proof let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume A1: -1<sn & sn<1 & q1`1>=0 & q2`1>=0 & |.q1.|<>0 & |.q2.|<>0
& q1`2/|.q1.|<q2`2/|.q2.|;
now per cases by A1;
case A2: q1`1>0;
now per cases by A1;
case q2`1>0;
hence (for p1,p2 being Point of TOP-REAL 2 st
p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2 holds
p1`2/|.p1.|<p2`2/|.p2.|) by A1,A2,JGRAPH_4:117;
case A3: q2`1=0;
then |.q2.|^2=(q2`2)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(q2`2)^2;
then A4: |.q2.|=q2`2 or |.q2.|=-(q2`2) by JGRAPH_3:1;
now assume |.q2.|=-(q2`2);
then 1=(-(q2`2))/|.q2.| by A1,XCMPLX_1:60;
then A5: q1`2/|.q1.|< -1 by A1,XCMPLX_1:191;
A6: |.q1.|>=0 by TOPRNS_1:26;
(|.q1.|)^2=(q1`2)^2+(q1`1)^2 by JGRAPH_3:10;
then (|.q1.|)^2-(q1`2)^2=(q1`1)^2 by XCMPLX_1:26;
then (|.q1.|)^2-(q1`2)^2>=0 by SQUARE_1:72;
then (|.q1.|)^2-(q1`2)^2+(q1`2)^2>=0+(q1`2)^2 by REAL_1:55;
then (|.q1.|)^2>=(q1`2)^2 by XCMPLX_1:27;
then -|.q1.|<=q1`2 & q1`2<=|.q1.| by A6,JGRAPH_2:5;
then (-|.q1.|)/|.q1.|<=q1`2/|.q1.| by A1,A6,REAL_1:73;
hence contradiction by A1,A5,XCMPLX_1:198;
end;
then A7: q2`2/|.q2.|=1 by A1,A4,XCMPLX_1:60;
thus for p1,p2 being Point of TOP-REAL 2 st
p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2 holds
p1`2/|.p1.|<p2`2/|.p2.|
proof let p1,p2 be Point of TOP-REAL 2;
assume A8: p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2;
then A9: p2=q2 by A3,JGRAPH_4:89;
A10: |.p1.|=|.q1.| by A8,JGRAPH_4:104;
A11: |.p1.|>=0 by TOPRNS_1:26;
(|.p1.|)^2=(p1`2)^2+(p1`1)^2 by JGRAPH_3:10;
then A12: (|.p1.|)^2-(p1`2)^2=(p1`1)^2 by XCMPLX_1:26;
then (|.p1.|)^2-(p1`2)^2>=0 by SQUARE_1:72;
then (|.p1.|)^2-(p1`2)^2+(p1`2)^2>=0+(p1`2)^2 by REAL_1:55;
then (|.p1.|)^2>=(p1`2)^2 by XCMPLX_1:27;
then -|.p1.|<=p1`2 & p1`2<=|.p1.| by A11,JGRAPH_2:5;
then (|.p1.|)/|.p1.|>=p1`2/|.p1.| by A1,A10,A11,REAL_1:73;
then A13: 1>= p1`2/|.p1.| by A1,A10,XCMPLX_1:60;
A14: p1`1>0 by A1,A2,A8,Th25;
now assume 1= p1`2/|.p1.|; then (1)*|.p1.|=p1`2 by A1,A10,XCMPLX_1:
88
;
then (|.p1.|)^2-(p1`2)^2=0 by XCMPLX_1:14;
hence contradiction by A12,A14,SQUARE_1:73;
end;
hence p1`2/|.p1.|<p2`2/|.p2.| by A7,A9,A13,REAL_1:def 5;
end;
end;
hence (for p1,p2 being Point of TOP-REAL 2 st
p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2 holds
p1`2/|.p1.|<p2`2/|.p2.|);
case A15: q1`1=0;
then |.q1.|^2=(q1`2)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(q1`2)^2;
then A16: |.q1.|=q1`2 or |.q1.|=-(q1`2) by JGRAPH_3:1;
now assume |.q1.|=(q1`2);
then A17: q2`2/|.q2.|> 1 by A1,XCMPLX_1:60;
A18: |.q2.|>=0 by TOPRNS_1:26;
(|.q2.|)^2=(q2`2)^2+(q2`1)^2 by JGRAPH_3:10;
then (|.q2.|)^2-(q2`2)^2=(q2`1)^2 by XCMPLX_1:26;
then (|.q2.|)^2-(q2`2)^2>=0 by SQUARE_1:72;
then (|.q2.|)^2-(q2`2)^2+(q2`2)^2>=0+(q2`2)^2 by REAL_1:55;
then (|.q2.|)^2>=(q2`2)^2 by XCMPLX_1:27;
then -|.q2.|<=q2`2 & q2`2<=|.q2.| by A18,JGRAPH_2:5;
then (|.q2.|)/|.q2.|>=q2`2/|.q2.| by A1,A18,REAL_1:73;
hence contradiction by A1,A17,XCMPLX_1:60;
end;
then (-(q1`2))/|.q1.|=1 by A1,A16,XCMPLX_1:60;
then A19: -(q1`2/|.q1.|)=1 by XCMPLX_1:188;
thus (for p1,p2 being Point of TOP-REAL 2 st
p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2
holds p1`2/|.p1.|<p2`2/|.p2.|)
proof let p1,p2 be Point of TOP-REAL 2;
assume A20: p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2;
then A21: p1=q1 by A15,JGRAPH_4:89;
A22: |.p2.|=|.q2.| by A20,JGRAPH_4:104;
A23: |.p2.|>=0 by TOPRNS_1:26;
(|.p2.|)^2=(p2`2)^2+(p2`1)^2 by JGRAPH_3:10;
then A24: (|.p2.|)^2-(p2`2)^2=(p2`1)^2 by XCMPLX_1:26;
then (|.p2.|)^2-(p2`2)^2>=0 by SQUARE_1:72;
then (|.p2.|)^2-(p2`2)^2+(p2`2)^2>=0+(p2`2)^2 by REAL_1:55;
then (|.p2.|)^2>=(p2`2)^2 by XCMPLX_1:27;
then -|.p2.|<=p2`2 & p2`2<=|.p2.| by A23,JGRAPH_2:5;
then (-|.p2.|)/|.p2.|<=p2`2/|.p2.| by A1,A22,A23,REAL_1:73;
then A25: -1 <= p2`2/|.p2.| by A1,A22,XCMPLX_1:198;
now per cases;
case q2`1=0;
then p2=q2 by A20,JGRAPH_4:89;
hence p2`2/|.p2.|>-1 by A1,A19;
case q2`1<>0;
then A26: p2`1>0 by A1,A20,Th25;
now assume -1= p2`2/|.p2.|; then (-1)*|.p2.|=p2`2 by A1,A22,XCMPLX_1
:88;
then (-|.p2.|)^2=(p2`2)^2 by XCMPLX_1:180;
then (|.p2.|)^2=(p2`2)^2 by SQUARE_1:61;
then (|.p2.|)^2-(p2`2)^2=0 by XCMPLX_1:14;
hence contradiction by A24,A26,SQUARE_1:73;
end;
hence
p2`2/|.p2.|>-1 by A25,REAL_1:def 5;
end;
hence p1`2/|.p1.|<p2`2/|.p2.| by A19,A21;
end;
end;
hence thesis;
end;
theorem Th28:
for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2<0
holds (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphS).q holds p`2<0)
proof let cn be Real,q be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q`2<0;
now per cases;
case q`1/|.q.|>=cn;
hence (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphS).q holds p`2<0) by A1,JGRAPH_4:144;
case q`1/|.q.|<cn;
hence (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphS).q holds p`2<0) by A1,JGRAPH_4:145;
end;
hence thesis;
end;
theorem Th29:
for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2<0
& q`1/|.q.|>cn holds (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphS).q holds p`2<0 & p`1>0)
proof let cn be Real,q be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q`2<0 & q`1/|.q.|>cn;
let p be Point of TOP-REAL 2;
assume A2: p=(cn-FanMorphS).q;
then A3: p`2<0 & p`1>=0 by A1,JGRAPH_4:144;
now assume A4: p`1=0;
then (|.p.|)^2=(p`2)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(p`2)^2;
then A5: p`2=|.p.| or p`2= - |.p.| by JGRAPH_3:1;
then A6: |.p.| <> 0 by A1,A2,JGRAPH_4:144;
set p1=(1/|.p.|)*p;
A7: |.p.|*p1=(|.p.|*(1/|.p.|))*p by EUCLID:34
.=(1)*p by A6,XCMPLX_1:107 .=p by EUCLID:33;
A8: p1=|[(1/|.p.|)*p`1,(1/|.p.|)*p`2]| by EUCLID:61;
then p1`2=(-|.p.|)*(1/|.p.|) by A3,A5,EUCLID:56,TOPRNS_1:26 .=-(|.p.|*(1/
|.p.|)) by XCMPLX_1:175
.=-1 by A6,XCMPLX_1:107;
then A9: p=|.p.|*(|[0,-1]|) by A4,A7,A8,EUCLID:56;
set q1=(|.p.|)*|[cn,-sqrt(1-cn^2)]|;
A10:(|[cn,-sqrt(1-cn^2)]|)`1=cn by EUCLID:56;
A11:(|[cn,-sqrt(1-cn^2)]|)`2=-sqrt(1-cn^2) by EUCLID:56;
then A12: q1=|[|.p.|*cn,|.p.|*(-sqrt(1-cn^2))]| by A10,EUCLID:61;
then A13: q1`1=(|.p.|)*cn by EUCLID:56;
A14: q1`2= (-sqrt(1-cn^2))*(|.p.|) by A12,EUCLID:56
.=-(sqrt(1-cn^2)*(|.p.|)) by XCMPLX_1:175;
A15: |.p.|>=0 by TOPRNS_1:26;
1^2>cn^2 by A1,JGRAPH_2:8;
then A16: 1-cn^2>0 by SQUARE_1:11,59;
then sqrt(1-cn^2)>0 by SQUARE_1:93;
then --sqrt(1-cn^2)*(|.p.|)>0 by A6,A15,SQUARE_1:21;
then A17: q1`2<0 by A14,REAL_1:66;
A18: |.q1.|=abs(|.p.|)*|.(|[cn,-sqrt(1-cn^2)]|).|
by TOPRNS_1:8
.=abs(|.p.|)*sqrt((cn)^2+(-sqrt(1-cn^2))^2)
by A10,A11,JGRAPH_3:10
.=abs(|.p.|)*sqrt(cn^2+(sqrt(1-cn^2))^2) by SQUARE_1:61
.=abs(|.p.|)*sqrt(cn^2+(1-cn^2)) by A16,SQUARE_1:def 4
.=abs(|.p.|)*1 by SQUARE_1:83,XCMPLX_1:27
.=|.p.| by A15,ABSVALUE:def 1;
then A19: q1`1/|.q1.|=cn by A6,A13,XCMPLX_1:90;
set p2=(cn-FanMorphS).q1;
A20: p2`2<0 & p2`1=0 by A1,A17,A19,JGRAPH_4:149;
then (|.p2.|)^2=(p2`2)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(p2`2)^2;
then A21: p2`2=|.p2.| or p2`2= - |.p2.| by JGRAPH_3:1;
|.p2.|=|.p.| by A18,JGRAPH_4:135;
then A22: p2=|[0,-(|.p.|)]| by A20,A21,EUCLID:57,TOPRNS_1:26;
(|[0,-1]|)`1=0 & (|[0,-1]|)`2=-1 by EUCLID:56;
then A23: |.p.|*(|[0,-1]|)=|[|.p.|*0,|.p.|*(-1)]| by EUCLID:61
.=|[0,-(|.p.|)]| by XCMPLX_1:180;
A24: (cn-FanMorphS) is one-to-one by A1,JGRAPH_4:140;
dom (cn-FanMorphS)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then q1=q by A2,A9,A22,A23,A24,FUNCT_1:def 8;
hence contradiction by A1,A6,A13,A18,XCMPLX_1:90;
end;
hence p`2<0 & p`1>0 by A1,A2,JGRAPH_4:144;
end;
theorem Th30:
for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & cn<1 & q1`2<=0
& q2`2<=0 & |.q1.|<>0 & |.q2.|<>0 & q1`1/|.q1.|<q2`1/|.q2.|
holds (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2 holds p1`1/|.p1.|<p2`1/|.p2.|)
proof let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q1`2<=0 & q2`2<=0 & |.q1.|<>0 & |.q2.|<>0
& q1`1/|.q1.|<q2`1/|.q2.|;
now per cases by A1;
case A2: q1`2<0;
now per cases by A1;
case q2`2<0;
hence (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2 holds
p1`1/|.p1.|<p2`1/|.p2.|) by A1,A2,JGRAPH_4:148;
case A3: q2`2=0;
then |.q2.|^2=(q2`1)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(q2`1)^2;
then A4: |.q2.|=q2`1 or |.q2.|=-(q2`1) by JGRAPH_3:1;
now assume |.q2.|=-(q2`1);
then 1=(-(q2`1))/|.q2.| by A1,XCMPLX_1:60;
then A5: q1`1/|.q1.|< -1 by A1,XCMPLX_1:191;
A6: |.q1.|>=0 by TOPRNS_1:26;
(|.q1.|)^2=(q1`1)^2+(q1`2)^2 by JGRAPH_3:10;
then (|.q1.|)^2-(q1`1)^2=(q1`2)^2 by XCMPLX_1:26;
then (|.q1.|)^2-(q1`1)^2>=0 by SQUARE_1:72;
then (|.q1.|)^2-(q1`1)^2+(q1`1)^2>=0+(q1`1)^2 by REAL_1:55;
then (|.q1.|)^2>=(q1`1)^2 by XCMPLX_1:27;
then -|.q1.|<=q1`1 & q1`1<=|.q1.| by A6,JGRAPH_2:5;
then (-|.q1.|)/|.q1.|<=q1`1/|.q1.| by A1,A6,REAL_1:73;
hence contradiction by A1,A5,XCMPLX_1:198;
end;
then A7: q2`1/|.q2.|=1 by A1,A4,XCMPLX_1:60;
thus for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2 holds
p1`1/|.p1.|<p2`1/|.p2.|
proof let p1,p2 be Point of TOP-REAL 2;
assume A8: p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2;
then A9: p2=q2 by A3,JGRAPH_4:120;
A10: |.p1.|=|.q1.| by A8,JGRAPH_4:135;
A11: |.p1.|>=0 by TOPRNS_1:26;
(|.p1.|)^2=(p1`1)^2+(p1`2)^2 by JGRAPH_3:10;
then A12: (|.p1.|)^2-(p1`1)^2=(p1`2)^2 by XCMPLX_1:26;
then (|.p1.|)^2-(p1`1)^2>=0 by SQUARE_1:72;
then (|.p1.|)^2-(p1`1)^2+(p1`1)^2>=0+(p1`1)^2 by REAL_1:55;
then (|.p1.|)^2>=(p1`1)^2 by XCMPLX_1:27;
then -|.p1.|<=p1`1 & p1`1<=|.p1.| by A11,JGRAPH_2:5;
then (|.p1.|)/|.p1.|>=p1`1/|.p1.| by A1,A10,A11,REAL_1:73;
then A13: 1>= p1`1/|.p1.| by A1,A10,XCMPLX_1:60;
A14: p1`2<0 by A1,A2,A8,Th28;
now assume 1= p1`1/|.p1.|; then (1)*|.p1.|=p1`1 by A1,A10,XCMPLX_1:
88
;
then (|.p1.|)^2-(p1`1)^2=0 by XCMPLX_1:14;
hence contradiction by A12,A14,SQUARE_1:73;
end;
hence p1`1/|.p1.|<p2`1/|.p2.| by A7,A9,A13,REAL_1:def 5;
end;
end;
hence (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2 holds
p1`1/|.p1.|<p2`1/|.p2.|);
case A15: q1`2=0;
then |.q1.|^2=(q1`1)^2+0 by JGRAPH_3:10,SQUARE_1:60 .=(q1`1)^2;
then A16: |.q1.|=q1`1 or |.q1.|=-(q1`1) by JGRAPH_3:1;
now assume |.q1.|=(q1`1);
then A17: q2`1/|.q2.|> 1 by A1,XCMPLX_1:60;
A18: |.q2.|>=0 by TOPRNS_1:26;
(|.q2.|)^2=(q2`1)^2+(q2`2)^2 by JGRAPH_3:10;
then (|.q2.|)^2-(q2`1)^2=(q2`2)^2 by XCMPLX_1:26;
then (|.q2.|)^2-(q2`1)^2>=0 by SQUARE_1:72;
then (|.q2.|)^2-(q2`1)^2+(q2`1)^2>=0+(q2`1)^2 by REAL_1:55;
then (|.q2.|)^2>=(q2`1)^2 by XCMPLX_1:27;
then -|.q2.|<=q2`1 & q2`1<=|.q2.| by A18,JGRAPH_2:5;
then (|.q2.|)/|.q2.|>=q2`1/|.q2.| by A1,A18,REAL_1:73;
hence contradiction by A1,A17,XCMPLX_1:60;
end;
then (-(q1`1))/|.q1.|=1 by A1,A16,XCMPLX_1:60;
then A19: -(q1`1/|.q1.|)=1 by XCMPLX_1:188;
thus (for p1,p2 being Point of TOP-REAL 2 st
p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2
holds p1`1/|.p1.|<p2`1/|.p2.|)
proof let p1,p2 be Point of TOP-REAL 2;
assume A20: p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2;
then A21: p1=q1 by A15,JGRAPH_4:120;
A22: |.p2.|=|.q2.| by A20,JGRAPH_4:135;
A23: |.p2.|>=0 by TOPRNS_1:26;
(|.p2.|)^2=(p2`1)^2+(p2`2)^2 by JGRAPH_3:10;
then A24: (|.p2.|)^2-(p2`1)^2=(p2`2)^2 by XCMPLX_1:26;
then (|.p2.|)^2-(p2`1)^2>=0 by SQUARE_1:72;
then (|.p2.|)^2-(p2`1)^2+(p2`1)^2>=0+(p2`1)^2 by REAL_1:55;
then (|.p2.|)^2>=(p2`1)^2 by XCMPLX_1:27;
then -|.p2.|<=p2`1 & p2`1<=|.p2.| by A23,JGRAPH_2:5;
then (-|.p2.|)/|.p2.|<=p2`1/|.p2.| by A1,A22,A23,REAL_1:73;
then A25: -1 <= p2`1/|.p2.| by A1,A22,XCMPLX_1:198;
now per cases;
case q2`2=0;
then p2=q2 by A20,JGRAPH_4:120;
hence p2`1/|.p2.|>-1 by A1,A19;
case q2`2<>0;
then A26: p2`2<0 by A1,A20,Th28;
now assume -1= p2`1/|.p2.|; then (-1)*|.p2.|=p2`1 by A1,A22,XCMPLX_1
:88;
then (-|.p2.|)^2=(p2`1)^2 by XCMPLX_1:180;
then (|.p2.|)^2=(p2`1)^2 by SQUARE_1:61;
then (|.p2.|)^2-(p2`1)^2=0 by XCMPLX_1:14;
hence contradiction by A24,A26,SQUARE_1:73;
end;
hence
p2`1/|.p2.|>-1 by A25,REAL_1:def 5;
end;
hence p1`1/|.p1.|<p2`1/|.p2.| by A19,A21;
end;
end;
hence thesis;
end;
begin :: Order of Points on Circle
Lm2: now let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={q: |.q.|=1};
A2: proj1.:P c= [.-1,1.]
proof let y be set;assume y in proj1.:P;
then consider x being set such that
A3: x in dom proj1 & x in P & y=proj1.x by FUNCT_1:def 12;
reconsider q=x as Point of TOP-REAL 2 by A3;
A4: y=q`1 by A3,PSCOMP_1:def 28;
consider q2 being Point of TOP-REAL 2 such that
A5: q2=x & |.q2.|=1 by A1,A3;
(q`1)^2+(q`2)^2=1 by A5,JGRAPH_3:10,SQUARE_1:59;
then A6: (q`2)^2=1-(q`1)^2 by XCMPLX_1:26;
0<=(q`2)^2 by SQUARE_1:72;
then 1-(q`1)^2+(q`1)^2 >=0+(q`1)^2 by A6,REAL_1:55;
then 1>=(q`1)^2 by XCMPLX_1:27;
then -1<=q`1 & q`1<=1 by JGRAPH_4:4;
hence y in [.-1,1.] by A4,TOPREAL5:1;
end;
[.-1,1.] c= proj1.:P
proof let y be set;assume
y in [.-1,1.];
then y in {r where r is Real: -1<=r & r<=1 } by RCOMP_1:def 1;
then consider r being Real such that
A7: y=r & -1<=r & r<=1;
set q=|[r,sqrt(1-r^2)]|;
A8: dom proj1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A9: q`1=r & q`2=sqrt(1-r^2) by EUCLID:56;
1^2>=r^2 by A7,JGRAPH_2:7;
then A10: 1-r^2>=0 by SQUARE_1:12,59;
|.q.|=sqrt(r^2+(sqrt(1-r^2))^2) by A9,JGRAPH_3:10
.=sqrt(r^2+(1-r^2)) by A10,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then A11: q in P by A1;
proj1.q=q`1 by PSCOMP_1:def 28 .=r by EUCLID:56;
hence y in proj1.:P by A7,A8,A11,FUNCT_1:def 12;
end;
hence proj1.:P=[.-1,1.] by A2,XBOOLE_0:def 10;
A12: proj2.:P c= [.-1,1.]
proof let y be set;assume
y in proj2.:P;
then consider x being set such that
A13: x in dom proj2 & x in P & y=proj2.x by FUNCT_1:def 12;
reconsider q=x as Point of TOP-REAL 2 by A13;
A14: y=q`2 by A13,PSCOMP_1:def 29;
consider q2 being Point of TOP-REAL 2 such that
A15: q2=x & |.q2.|=1 by A1,A13;
(q`1)^2+(q`2)^2=1 by A15,JGRAPH_3:10,SQUARE_1:59;
then A16: (q`1)^2=1-(q`2)^2 by XCMPLX_1:26;
0<=(q`1)^2 by SQUARE_1:72;
then 1-(q`2)^2+(q`2)^2 >=0+(q`2)^2 by A16,REAL_1:55;
then 1>=(q`2)^2 by XCMPLX_1:27;
then -1<=q`2 & q`2<=1 by JGRAPH_4:4;
hence y in [.-1,1.] by A14,TOPREAL5:1;
end;
[.-1,1.] c= proj2.:P
proof let y be set;assume
y in [.-1,1.];
then y in {r where r is Real: -1<=r & r<=1 } by RCOMP_1:def 1;
then consider r being Real such that
A17: y=r & -1<=r & r<=1;
set q=|[sqrt(1-r^2),r]|;
A18: dom proj2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A19: q`2=r & q`1=sqrt(1-r^2) by EUCLID:56;
1^2>=r^2 by A17,JGRAPH_2:7;
then A20: 1-r^2>=0 by SQUARE_1:12,59;
|.q.|=sqrt((sqrt(1-r^2))^2+r^2) by A19,JGRAPH_3:10
.=sqrt((1-r^2)+r^2) by A20,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then A21: q in P by A1;
proj2.q=q`2 by PSCOMP_1:def 29 .=r by EUCLID:56;
hence y in proj2.:P by A17,A18,A21,FUNCT_1:def 12;
end;
hence proj2.:P=[.-1,1.] by A12,XBOOLE_0:def 10;
end;
Lm3: for P being compact non empty Subset of TOP-REAL 2 st
P={q: |.q.|=1} holds W-bound(P)=-1
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={q: |.q.|=1};
A2: the carrier of ((TOP-REAL 2)|P) = P by JORDAN1:1;
proj1.:P=[.-1,1.] by A1,Lm2;
then (proj1|P).:P=[.-1,1.] by RFUNCT_2:5;
then (proj1||P).:P=[.-1,1.] by PSCOMP_1:def 26;
then inf ((proj1||P).:P)=-1 by JORDAN5A:20;
then inf (proj1||P)=-1 by A2,PSCOMP_1:def 20;
hence W-bound P=-1 by PSCOMP_1:def 30;
end;
theorem Th31: for P being compact non empty Subset of TOP-REAL 2 st
P={q: |.q.|=1}
holds W-bound(P)=-1 & E-bound(P)=1 & S-bound(P)=-1 & N-bound(P)=1
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={q: |.q.|=1};
hence W-bound(P)=-1 by Lm3;
A2: the carrier of ((TOP-REAL 2)|P) =P by JORDAN1:1;
proj1.:P=[.-1,1.] by A1,Lm2;
then (proj1|P).:P=[.-1,1.] by RFUNCT_2:5;
then (proj1||P).:P=[.-1,1.] by PSCOMP_1:def 26;
then sup ((proj1||P).:the carrier of ((TOP-REAL 2)|P))=1 by A2,JORDAN5A:20;
then sup (proj1||P)=1 by PSCOMP_1:def 21;
hence E-bound P=1 by PSCOMP_1:def 32;
proj2.:P=[.-1,1.] by A1,Lm2;
then (proj2|P).:P=[.-1,1.] by RFUNCT_2:5;
then A3: (proj2||P).:P=[.-1,1.] by PSCOMP_1:def 26;
then inf ((proj2||P).:P)=-1 by JORDAN5A:20;
then inf (proj2||P)=-1 by A2,PSCOMP_1:def 20;
hence S-bound P=-1 by PSCOMP_1:def 33;
sup ((proj2||P).:P)=1 by A3,JORDAN5A:20;
then sup (proj2||P)=1 by A2,PSCOMP_1:def 21;
hence N-bound P=1 by PSCOMP_1:def 31;
end;
theorem Th32: for P being compact non empty Subset of TOP-REAL 2 st
P={q: |.q.|=1}
holds W-min(P)=|[-1,0]|
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={q: |.q.|=1};
A2: the carrier of ((TOP-REAL 2)|P) = P by JORDAN1:1;
A3: W-bound P=-1 by A1,Lm3;
proj2.:P=[.-1,1.] by A1,Lm2;
then (proj2|P).:P=[.-1,1.] by RFUNCT_2:5;
then A4: (proj2||P).:P=[.-1,1.] by PSCOMP_1:def 26;
then inf ((proj2||P).:P)=-1 by JORDAN5A:20;
then inf (proj2||P)=-1 by A2,PSCOMP_1:def 20;
then S-bound P=-1 by PSCOMP_1:def 33;
then A5: SW-corner P=|[-1,-1]| by A3,PSCOMP_1:def 34;
sup ((proj2||P).:P)=1 by A4,JORDAN5A:20;
then sup (proj2||P)=1 by A2,PSCOMP_1:def 21;
then N-bound P=1 by PSCOMP_1:def 31;
then A6: NW-corner P=|[-1,1]| by A3,PSCOMP_1:def 35;
A7: {|[-1,0]|} c= LSeg(SW-corner P, NW-corner P)/\P
proof let x be set;assume
x in {|[-1,0]|};
then A8: x=|[-1,0]| by TARSKI:def 1;
set q=|[-1,0]|;
q`2=0 & q`1=-1 by EUCLID:56;
then |.q.|=sqrt((-1)^2+0^2) by JGRAPH_3:10
.=1 by SQUARE_1:59,60,61,83;
then A9: x in P by A1,A8;
q=|[(1/2)*(-1)+(1/2)*(-1),(1/2)*(-1)+(1/2)*1]|;
then q=|[(1/2)*(-1),(1/2)*(-1)]|+|[(1/2)*(-1),(1/2)*1]| by EUCLID:60;
then q=|[(1/2)*(-1),(1/2)*(-1)]|+(1/2)*|[-1,1]| by EUCLID:62;
then q=(1/2)*|[-1,-1]|+(1-(1/2))*|[-1,1]| by EUCLID:62;
then q in { (1-l)*(SW-corner P) + l*(NW-corner P) where l is Real:
0 <= l & l <= 1 } by A5,A6;
then x in LSeg(SW-corner P, NW-corner P) by A8,TOPREAL1:def 4;
hence x in LSeg(SW-corner P, NW-corner P)/\P by A9,XBOOLE_0:def 3;
end;
LSeg(SW-corner P, NW-corner P)/\P c= {|[-1,0]|}
proof let x be set;assume x in LSeg(SW-corner P, NW-corner P)/\P;
then A10: x in LSeg(SW-corner P, NW-corner P) & x in P by XBOOLE_0:def 3;
then x in { (1-l)*(SW-corner P) + l*(NW-corner P) where l is Real:
0 <= l & l <= 1 } by TOPREAL1:def 4;
then consider l being Real such that
A11: x=(1-l)*(SW-corner P)+l*(NW-corner P)
& 0<=l & l<=1;
x=|[(1-l)*(-1),(1-l)*(-1)]|+(l)*|[-1,1]| by A5,A6,A11,EUCLID:62;
then x=|[(1-l)*(-1),(1-l)*(-1)]|+|[(l)*(-1),(l)*1]| by EUCLID:62;
then x=|[(1-l)*(-1)+(l)*(-1),(1-l)*(-1)+(l)*1]| by EUCLID:60;
then x=|[((1-l)+l)*(-1),(1-l)*(-1)+(l)*1]| by XCMPLX_1:8;
then A12: x=|[(1)*(-1),(1-l)*(-1)+(l)*1]| by XCMPLX_1:27;
reconsider q3=x as Point of TOP-REAL 2 by A11;
A13: q3`1=-1 & q3`2=(1-l)*(-1)+l by A12,EUCLID:56;
consider q2 being Point of TOP-REAL 2 such that
A14: q2=x & |.q2.|=1 by A1,A10;
A15: 1=sqrt((-1)^2+(q3`2)^2) by A13,A14,JGRAPH_3:10
.=sqrt(1+(q3`2)^2) by SQUARE_1:59,61;
now assume (q3`2)^2>0; then 1<1+(q3`2)^2 by REAL_1:69;
hence contradiction by A15,SQUARE_1:83,95;
end;
then (q3`2)^2=0 by SQUARE_1:72;
then q3`2=0 by SQUARE_1:73;
hence x in {|[-1,0]|} by A12,A13,TARSKI:def 1;
end;
then LSeg(SW-corner P, NW-corner P)/\P={|[-1,0]|} by A7,XBOOLE_0:def 10;
then A16: W-most P={|[-1,0]|} by PSCOMP_1:def 38;
(proj2||W-most P).:the carrier of ((TOP-REAL 2)|(W-most P))
=(proj2||W-most P).:(W-most P) by JORDAN1:1
.=(proj2|(W-most P)).:(W-most P) by PSCOMP_1:def 26
.=proj2.:(W-most P) by RFUNCT_2:5
.={proj2.(|[-1,0]|)} by A16,SETWISEO:13
.={(|[-1,0]|)`2} by PSCOMP_1:def 29
.={0} by EUCLID:56;
then inf ((proj2||W-most P).:the carrier of ((TOP-REAL 2)|(W-most P)))
=0 by SEQ_4:22;
then inf (proj2||W-most P)=0 by PSCOMP_1:def 20;
hence W-min(P)=|[-1,0]| by A3,PSCOMP_1:def 42;
end;
theorem Th33: for P being compact non empty Subset of TOP-REAL 2 st
P={q: |.q.|=1}
holds E-max(P)=|[1,0]|
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={q: |.q.|=1};
A2: the carrier of ((TOP-REAL 2)|P) =P by JORDAN1:1;
A3: E-bound P=1 by A1,Th31;
proj2.:P=[.-1,1.] by A1,Lm2;
then (proj2|P).:P=[.-1,1.] by RFUNCT_2:5;
then A4: (proj2||P).:P=[.-1,1.] by PSCOMP_1:def 26;
then inf ((proj2||P).:P)=-1 by JORDAN5A:20;
then inf (proj2||P)=-1 by A2,PSCOMP_1:def 20;
then S-bound P=-1 by PSCOMP_1:def 33;
then A5: SE-corner P=|[1,-1]| by A3,PSCOMP_1:def 37;
sup ((proj2||P).:P)=1 by A4,JORDAN5A:20;
then sup (proj2||P)=1 by A2,PSCOMP_1:def 21;
then N-bound P=1 by PSCOMP_1:def 31;
then A6: NE-corner P=|[1,1]| by A3,PSCOMP_1:def 36;
A7: {|[1,0]|} c= LSeg(SE-corner P, NE-corner P)/\P
proof let x be set;assume
x in {|[1,0]|};
then A8: x=|[1,0]| by TARSKI:def 1;
set q=|[1,0]|;
q`2=0 & q`1=1 by EUCLID:56;
then |.q.|=sqrt((1)^2+0^2) by JGRAPH_3:10
.=1 by SQUARE_1:59,60,83;
then A9: x in P by A1,A8;
q=|[(1/2)*(1)+(1/2)*(1),(1/2)*(-1)+(1/2)*1]|;
then q=|[(1/2)*(1),(1/2)*(-1)]|+|[(1/2)*(1),(1/2)*1]| by EUCLID:60;
then q=|[(1/2)*(1),(1/2)*(-1)]|+(1/2)*|[1,1]| by EUCLID:62;
then q=(1/2)*|[1,-1]|+(1-(1/2))*|[1,1]| by EUCLID:62;
then q in { (1-l)*(SE-corner P) + l*(NE-corner P) where l is Real:
0 <= l & l <= 1 } by A5,A6;
then x in LSeg(SE-corner P, NE-corner P) by A8,TOPREAL1:def 4;
hence x in LSeg(SE-corner P, NE-corner P)/\P by A9,XBOOLE_0:def 3;
end;
LSeg(SE-corner P, NE-corner P)/\P c= {|[1,0]|}
proof let x be set;assume x in LSeg(SE-corner P, NE-corner P)/\P;
then A10: x in LSeg(SE-corner P, NE-corner P) & x in P by XBOOLE_0:def 3;
then x in { (1-l)*(SE-corner P) + l*(NE-corner P) where l is Real:
0 <= l & l <= 1 } by TOPREAL1:def 4;
then consider l being Real such that
A11: x=(1-l)*(SE-corner P)+l*(NE-corner P)
& 0<=l & l<=1;
x=|[(1-l)*(1),(1-l)*(-1)]|+(l)*|[1,1]| by A5,A6,A11,EUCLID:62;
then x=|[(1-l)*(1),(1-l)*(-1)]|+|[(l)*(1),(l)*1]| by EUCLID:62;
then x=|[((1-l)+l)*(1),(1-l)*(-1)+(l)*1]| by EUCLID:60;
then A12: x=|[1,(1-l)*(-1)+l]| by XCMPLX_1:27;
reconsider q3=x as Point of TOP-REAL 2 by A11;
A13: q3`1=1 & q3`2=(1-l)*(-1)+l by A12,EUCLID:56;
consider q2 being Point of TOP-REAL 2 such that
A14: q2=x & |.q2.|=1 by A1,A10;
now assume (q3`2)^2>0; then 1<1+(q3`2)^2 by REAL_1:69;
hence contradiction by A13,A14,JGRAPH_3:10,SQUARE_1:59;
end;
then (q3`2)^2=0 by SQUARE_1:72;
then q3`2=0 by SQUARE_1:73;
hence x in {|[1,0]|} by A12,A13,TARSKI:def 1;
end;
then LSeg(SE-corner P, NE-corner P)/\P={|[1,0]|} by A7,XBOOLE_0:def 10;
then A15: E-most P={|[1,0]|} by PSCOMP_1:def 40;
(proj2||E-most P).:the carrier of ((TOP-REAL 2)|(E-most P))
=(proj2||E-most P).:(E-most P) by JORDAN1:1
.=(proj2|(E-most P)).:(E-most P) by PSCOMP_1:def 26
.=proj2.:(E-most P) by RFUNCT_2:5
.={proj2.(|[1,0]|)} by A15,SETWISEO:13
.={(|[1,0]|)`2} by PSCOMP_1:def 29
.={0} by EUCLID:56;
then sup ((proj2||E-most P).:the carrier of ((TOP-REAL 2)|(E-most P)))
=0 by SEQ_4:22;
then sup (proj2||E-most P)=0 by PSCOMP_1:def 21;
hence E-max(P)=|[1,0]| by A3,PSCOMP_1:def 46;
end;
theorem
for f being map of TOP-REAL 2,R^1 st
(for p being Point of TOP-REAL 2 holds f.p=proj1.p) holds f is continuous
proof let f be map of TOP-REAL 2,R^1;
assume A1:for p being Point of TOP-REAL 2 holds f.p=proj1.p;
(TOP-REAL 2)|([#](TOP-REAL 2))=TOP-REAL 2 by TSEP_1:3;
hence f is continuous by A1,JGRAPH_2:39;
end;
theorem Th35:
for f being map of TOP-REAL 2,R^1 st
(for p being Point of TOP-REAL 2 holds f.p=proj2.p) holds f is continuous
proof let f be map of TOP-REAL 2,R^1;
assume A1:for p being Point of TOP-REAL 2 holds f.p=proj2.p;
(TOP-REAL 2)|([#](TOP-REAL 2))=TOP-REAL 2 by TSEP_1:3;
hence f is continuous by A1,JGRAPH_2:40;
end;
theorem Th36: for P being compact non empty Subset of TOP-REAL 2 st
P={q where q is Point of TOP-REAL 2: |.q.|=1} holds
Upper_Arc(P) c= P & Lower_Arc(P) c= P
proof let P be compact non empty Subset of TOP-REAL 2;
assume P={q where q is Point of TOP-REAL 2: |.q.|=1};
then P is_simple_closed_curve by JGRAPH_3:36;
hence thesis by JORDAN1A:16;
end;
theorem Th37: for P being compact non empty Subset of TOP-REAL 2 st
P={q where q is Point of TOP-REAL 2: |.q.|=1} holds
Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={q where q is Point of TOP-REAL 2: |.q.|=1};
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
A4: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A2,JORDAN6:def 9;
consider P2 being non empty Subset of TOP-REAL 2
such that
A5: P2 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P2={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P2=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P2,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 8;
set P4=Lower_Arc(P);
A6: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
A7: W-bound(P)=-1 by A1,Th31;
A8: E-bound(P)=1 by A1,Th31;
A9: Vertical_Line(0)={p where p is Point of TOP-REAL 2: p`1=0}
by JORDAN6:def 6;
set P1=Upper_Arc(P), P2=Lower_Arc(P), Q=Vertical_Line(0);
set p11=W-min(P), p22=E-max(P);
set p8= First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line(0));
set pj= Last_Point(Lower_Arc(P),E-max(P),W-min(P),
Vertical_Line(0));
A10: W-bound P=-1 by A1,Th31;
A11: E-bound P=1 by A1,Th31;
A12: S-bound P=-1 by A1,Th31;
A13: N-bound P=1 by A1,Th31;
then A14: LSeg(|[0,-1]|,|[0,1]|) meets Upper_Arc P
by A2,A10,A11,A12,JORDAN1B:26;
A15: LSeg(|[0,-1]|,|[0,1]|) c= Q
proof let x be set;assume x in LSeg(|[0,-1]|,|[0,1]|);
then x in {(1-l)*(|[0,-1]|) +l*(|[0,1]|) where l is Real: 0<=l & l
<=1}
by TOPREAL1:def 4;
then consider l being Real such that
A16: x=(1-l)*(|[0,-1]|) +l*(|[0,1]|) & 0<=l & l<=1;
((1-l)*(|[0,-1]|) +l*(|[0,1]|))`1
= ((1-l)*(|[0,-1]|))`1 +(l*(|[0,1]|))`1 by TOPREAL3:7
.=(1-l)*(|[0,-1]|)`1+(l*(|[0,1]|))`1 by TOPREAL3:9
.=(1-l)*(|[0,-1]|)`1+l*((|[0,1]|))`1 by TOPREAL3:9
.=(1-l)*0+l*((|[0,1]|))`1 by EUCLID:56
.=(1-l)*0+l*0 by EUCLID:56
.=0;
hence x in Q by A9,A16;
end;
then A17: P1 meets Q by A14,XBOOLE_1:64;
A18: Upper_Arc(P) is closed by A3,JORDAN6:12;
Vertical_Line(0) is closed by JORDAN6:33;
then P1 /\ Q is closed by A18,TOPS_1:35;
then A19: p8 in P1 /\ Q &
for g being map of I[01], (TOP-REAL 2)|P1, s2 being Real st
g is_homeomorphism & g.0 = p11 & g.1 = p22
& g.s2 = p8 & 0 <= s2 & s2 <= 1 holds
(for t being Real st 0 <= t & t < s2 holds not g.t in Q)
by A3,A17,JORDAN5C:def 1;
P1 /\ Q c= {|[0,-1]|,|[0,1]|}
proof let x be set;assume x in P1 /\ Q;
then A20: x in P1 & x in Q by XBOOLE_0:def 3;
then consider p being Point of TOP-REAL 2 such that
A21: p=x & p`1=0 by A9;
x in P by A5,A20,XBOOLE_0:def 2;
then consider q being Point of TOP-REAL 2 such that
A22: q=x & |.q.|=1 by A1;
0+(q`2)^2 =1 by A21,A22,JGRAPH_3:10,SQUARE_1:59,60;
then q`2=1 or q`2=-1 by JGRAPH_3:2;
then x=|[0,-1]| or x=|[0,1]| by A21,A22,EUCLID:57;
hence x in {|[0,-1]|,|[0,1]|} by TARSKI:def 2;
end;
then p8=|[0,-1]| or p8=|[0,1]| by A19,TARSKI:def 2;
then A23: p8`2=-1 or p8`2=1 by EUCLID:56;
LSeg(|[0,-1]|,|[0,1]|) meets Lower_Arc P
by A2,A10,A11,A12,A13,JORDAN1B:27;
then A24: P2 meets Q by A15,XBOOLE_1:64;
A25: Lower_Arc(P) is closed by A4,JORDAN6:12;
Vertical_Line(0) is closed by JORDAN6:33;
then P2 /\ Q is closed by A25,TOPS_1:35;
then A26: pj in P2 /\ Q &
for g being map of I[01], (TOP-REAL 2)|P2, s2 being Real st
g is_homeomorphism & g.0 = p22 & g.1 = p11
& g.s2 = pj & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds not g.t in Q
by A4,A24,JORDAN5C:def 2;
P2 /\ Q c= {|[0,-1]|,|[0,1]|}
proof let x be set;assume x in P2 /\ Q;
then A27: x in P2 & x in Q by XBOOLE_0:def 3;
then consider p being Point of TOP-REAL 2 such that
A28: p=x & p`1=0 by A9;
x in P by A6,A27,XBOOLE_0:def 2;
then consider q being Point of TOP-REAL 2 such that
A29: q=x & |.q.|=1 by A1;
0+(q`2)^2 =1 by A28,A29,JGRAPH_3:10,SQUARE_1:59,60;
then q`2=1 or q`2=-1 by JGRAPH_3:2;
then x=|[0,-1]| or x=|[0,1]| by A28,A29,EUCLID:57;
hence x in {|[0,-1]|,|[0,1]|} by TARSKI:def 2;
end;
then pj=|[0,-1]| or pj=|[0,1]| by A26,TARSKI:def 2;
then A30: pj`2=-1 or pj`2=1 by EUCLID:56;
A31: p8 in P1 by A19,XBOOLE_0:def 3;
A32: Upper_Arc(P) c= P by A5,XBOOLE_1:7;
A33: Lower_Arc(P) c= P by A6,XBOOLE_1:7;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A34: E-max(P) in Upper_Arc(P) by A5,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A35: W-min(P) in Upper_Arc(P) by A5,XBOOLE_0:def 3;
reconsider R=Upper_Arc(P) as non empty Subset of TOP-REAL 2;
consider f being map of I[01], (TOP-REAL 2)|R such that
A36: f is_homeomorphism & f.0 =W-min(P) &
f.1 =E-max(P) by A3,TOPREAL1:def 2;
rng f =[#]((TOP-REAL 2)|R) by A36,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x8 being set such that
A37: x8 in dom f & p8=f.x8 by A31,FUNCT_1:def 5;
dom f= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x8 in {r where r is Real: 0<=r & r<=1 } by A37,RCOMP_1:def 1;
then consider r8 being Real such that
A38: x8=r8 & 0<=r8 & r8<=1;
A39: now assume r8=0;
then p8=|[-1,0]| by A1,A36,A37,A38,Th32;
hence contradiction by A23,EUCLID:56;
end;
now assume r8=1;
then p8=|[1,0]| by A1,A36,A37,A38,Th33;
hence contradiction by A23,EUCLID:56;
end;
then A40: 1>r8 by A38,REAL_1:def 5;
reconsider h2=proj2 as map of TOP-REAL 2,R^1 by JGRAPH_2:17;
A41: f is continuous by A36,TOPS_2:def 5;
A42: f is one-to-one by A36,TOPS_2:def 5;
for p being Point of (TOP-REAL 2) holds h2.p=proj2.p;
then A43: h2 is continuous by Th35;
A44: dom f=the carrier of I[01] by FUNCT_2:def 1;
A45: the carrier of ((TOP-REAL 2)|R)=R by JORDAN1:1;
then A46: rng f c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
dom h2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then A47: dom (h2*f)=the carrier of I[01] by A44,A46,RELAT_1:46;
rng (h2*f) c= rng h2 by RELAT_1:45;
then rng (h2*f) c= the carrier of R^1 by XBOOLE_1:1;
then h2*f is Function of the carrier of I[01],the carrier of R^1
by A47,FUNCT_2:4;
then reconsider g0=h2*f as map of I[01],R^1;
A48: (ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f.t=p & p`2>0) implies for q being Point of TOP-REAL 2
st q in Upper_Arc(P) holds q`2>=0
proof assume
(ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f.t=p & p`2>0);
then consider p being Point of TOP-REAL 2,t being Real such that
A49: 0<t & t<1 & f.t=p & p`2>0;
now assume ex q being Point of TOP-REAL 2
st q in Upper_Arc(P) & q`2<0;
then consider q being Point of TOP-REAL 2 such that
A50: q in Upper_Arc(P) & q`2<0;
rng f =[#]((TOP-REAL 2)|R) by A36,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x being set such that
A51: x in dom f & q=f.x by A50,FUNCT_1:def 5;
A52: dom f= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x in {r where r is Real: 0<=r & r<=1 } by A51,RCOMP_1:def 1;
then consider r being Real such that
A53: x=r & 0<=r & r<=1;
A54: (h2*f).r=h2.q by A51,A53,FUNCT_1:23
.=q`2 by PSCOMP_1:def 29;
t in {v where v is Real: 0<=v & v<=1 } by A49;
then A55: t in [.0,1.] by RCOMP_1:def 1;
then A56: (h2*f).t=h2.p by A49,A52,FUNCT_1:23
.=p`2 by PSCOMP_1:def 29;
now per cases by REAL_1:def 5;
case A57: r<t;
[.r,t.] c= [.0,1.] by A51,A52,A53,A55,RCOMP_1:16;
then reconsider B=[.r,t.] as non empty Subset of I[01]
by A57,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g0 is continuous by A41,A43,Th10;
then A58: g is continuous by TOPMETR:10;
A59: Closed-Interval-TSpace(r,t)=I[01]|B by A49,A53,A57,Th6,TOPMETR:27;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A57;
then A60: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A57;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A49,A50,A54,A56,A60,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A61: g.r1=0 & r<r1 & r1<t by A57,A58,A59,TOPREAL5:14;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A61;
then A62: r1 in B by RCOMP_1:def 1;
A63: 0<r1 by A53,A61;
r1<1 by A49,A61,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A63;
then A64: r1 in dom f by A52,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 5;
then f.r1 in R by A45;
then f.r1 in P by A32;
then consider q3 being Point of TOP-REAL 2 such that
A65: q3=f.r1 & |.q3.|=1 by A1;
A66: q3`2=h2.(f.r1) by A65,PSCOMP_1:def 29
.=g0.r1 by A64,FUNCT_1:23
.=0 by A61,A62,FUNCT_1:72;
then A67: 1=(q3`1)^2 +0^2 by A65,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A67,JGRAPH_3:2;
case q3`1=1;
then A68: q3=|[1,0]| by A66,EUCLID:57
.=E-max(P) by A1,Th33;
1 in dom f by A52,RCOMP_1:15;
hence contradiction
by A36,A42,A49,A61,A64,A65,A68,FUNCT_1:def 8;
case q3`1=-1;
then A69: q3=|[-1,0]| by A66,EUCLID:57
.=W-min(P) by A1,Th32;
0 in dom f by A52,RCOMP_1:15;
hence contradiction
by A36,A42,A53,A61,A64,A65,A69,FUNCT_1:def 8;
end;
hence contradiction;
case A70: t<r;
[.t,r.] c= [.0,1.] by A51,A52,A53,A55,RCOMP_1:16;
then reconsider B=[.t,r.] as non empty Subset of I[01]
by A70,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g0 is continuous by A41,A43,Th10;
then A71: g is continuous by TOPMETR:10;
A72: Closed-Interval-TSpace(t,r)=I[01]|B by A49,A53,A70,Th6,TOPMETR:27;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A70;
then A73: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A70;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A49,A50,A54,A56,A73,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A74: g.r1=0 & t<r1 & r1<r by A70,A71,A72,TOPREAL5:14;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A74;
then A75: r1 in B by RCOMP_1:def 1;
A76: 0<r1 by A49,A74,AXIOMS:22;
r1<1 by A53,A74,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A76;
then A77: r1 in dom f by A52,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 5;
then f.r1 in R by A45;
then f.r1 in P by A32;
then consider q3 being Point of TOP-REAL 2 such that
A78: q3=f.r1 & |.q3.|=1 by A1;
A79: q3`2=h2.(f.r1) by A78,PSCOMP_1:def 29
.=(h2*f).r1 by A77,FUNCT_1:23
.=0 by A74,A75,FUNCT_1:72;
then A80: 1=(q3`1)^2 +0^2 by A78,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A80,JGRAPH_3:2;
case q3`1=1;
then A81: q3=|[1,0]| by A79,EUCLID:57
.=E-max(P) by A1,Th33;
1 in dom f by A52,RCOMP_1:15;
hence contradiction
by A36,A42,A53,A74,A77,A78,A81,FUNCT_1:def 8;
case q3`1=-1;
then A82: q3=|[-1,0]| by A79,EUCLID:57
.=W-min(P) by A1,Th32;
0 in dom f by A52,RCOMP_1:15;
hence contradiction
by A36,A42,A49,A74,A77,A78,A82,FUNCT_1:def 8;
end;
hence contradiction;
case t=r;
hence contradiction by A49,A50,A54,A56;
end;
hence contradiction;
end;
hence for q being Point of TOP-REAL 2
st q in Upper_Arc(P) holds q`2>=0;
end;
reconsider R=Lower_Arc(P) as non empty Subset of TOP-REAL 2;
consider f2 being map of I[01], (TOP-REAL 2)|R such that
A83: f2 is_homeomorphism & f2.0 =E-max(P) &
f2.1 =W-min(P) by A4,TOPREAL1:def 2;
A84: f2 is continuous by A83,TOPS_2:def 5;
A85: f2 is one-to-one by A83,TOPS_2:def 5;
for p being Point of (TOP-REAL 2) holds
h2.p=proj2.p;
then A86: h2 is continuous by Th35;
A87: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A88: the carrier of ((TOP-REAL 2)|R)=R by JORDAN1:1;
then A89: rng f2 c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
dom h2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then A90: dom (h2*f2)=the carrier of I[01] by A87,A89,RELAT_1:46;
rng (h2*f2) c= rng h2 by RELAT_1:45;
then rng (h2*f2) c= the carrier of R^1 by XBOOLE_1:1;
then h2*f2 is Function of the carrier of I[01],the carrier of R^1
by A90,FUNCT_2:4;
then reconsider g1=h2*f2 as map of I[01],R^1;
A91: (ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f2.t=p & p`2>0) implies for q being Point of TOP-REAL 2
st q in Lower_Arc(P) holds q`2>=0
proof assume
(ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f2.t=p & p`2>0);
then consider p being Point of TOP-REAL 2,t being Real such that
A92: 0<t & t<1 & f2.t=p & p`2>0;
now assume ex q being Point of TOP-REAL 2
st q in Lower_Arc(P) & q`2<0;
then consider q being Point of TOP-REAL 2 such that
A93: q in Lower_Arc(P) & q`2<0;
rng f2 =[#]((TOP-REAL 2)|R) by A83,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x being set such that
A94: x in dom f2 & q=f2.x by A93,FUNCT_1:def 5;
A95: dom f2= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x in {r where r is Real: 0<=r & r<=1 } by A94,RCOMP_1:def 1;
then consider r being Real such that
A96: x=r & 0<=r & r<=1;
A97: (h2*f2).r=h2.q by A94,A96,FUNCT_1:23
.=q`2 by PSCOMP_1:def 29;
t in {v where v is Real: 0<=v & v<=1 } by A92;
then A98: t in [.0,1.] by RCOMP_1:def 1;
then A99: (h2*f2).t=h2.p by A92,A95,FUNCT_1:23
.=p`2 by PSCOMP_1:def 29;
now per cases by REAL_1:def 5;
case A100: r<t;
[.r,t.] c= [.0,1.] by A94,A95,A96,A98,RCOMP_1:16;
then reconsider B=[.r,t.] as non empty Subset of I[01]
by A100,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g1 is continuous by A84,A86,Th10;
then A101: g is continuous by TOPMETR:10;
A102: Closed-Interval-TSpace(r,t)=I[01]|B by A92,A96,A100,Th6,TOPMETR:27
;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A100;
then A103: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A100;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A92,A93,A97,A99,A103,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A104: g.r1=0 & r<r1 & r1<t by A100,A101,A102,TOPREAL5:14;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A104;
then A105: r1 in B by RCOMP_1:def 1;
A106: 0<r1 by A96,A104;
r1<1 by A92,A104,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A106;
then A107: r1 in dom f2 by A95,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 5;
then f2.r1 in R by A88;
then f2.r1 in P by A33;
then consider q3 being Point of TOP-REAL 2 such that
A108: q3=f2.r1 & |.q3.|=1 by A1;
A109: q3`2=h2.(f2.r1) by A108,PSCOMP_1:def 29
.=(h2*f2).r1 by A107,FUNCT_1:23
.=0 by A104,A105,FUNCT_1:72;
then A110: 1=(q3`1)^2 +0^2 by A108,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A110,JGRAPH_3:2;
case q3`1=1;
then A111: q3=|[1,0]| by A109,EUCLID:57
.=E-max(P) by A1,Th33;
0 in dom f2 by A95,RCOMP_1:15;
hence contradiction
by A83,A85,A96,A104,A107,A108,A111,FUNCT_1:def 8;
case q3`1=-1;
then A112: q3=|[-1,0]| by A109,EUCLID:57
.=W-min(P) by A1,Th32;
1 in dom f2 by A95,RCOMP_1:15;
hence contradiction
by A83,A85,A92,A104,A107,A108,A112,FUNCT_1:def 8;
end;
hence contradiction;
case A113: t<r;
[.t,r.] c= [.0,1.] by A94,A95,A96,A98,RCOMP_1:16;
then reconsider B=[.t,r.] as non empty Subset of I[01]
by A113,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g1 is continuous by A84,A86,Th10;
then A114: g is continuous by TOPMETR:10;
A115: Closed-Interval-TSpace(t,r)=I[01]|B by A92,A96,A113,Th6,TOPMETR:27
;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A113;
then A116: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A113;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A92,A93,A97,A99,A116,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A117: g.r1=0 & t<r1 & r1<r by A113,A114,A115,TOPREAL5:14;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A117;
then A118: r1 in B by RCOMP_1:def 1;
A119: 0<r1 by A92,A117,AXIOMS:22;
r1<1 by A96,A117,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A119;
then A120: r1 in dom f2 by A95,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 5;
then f2.r1 in R by A88;
then f2.r1 in P by A33;
then consider q3 being Point of TOP-REAL 2 such that
A121: q3=f2.r1 & |.q3.|=1 by A1;
A122: q3`2=h2.(f2.r1) by A121,PSCOMP_1:def 29
.=g1.r1 by A120,FUNCT_1:23
.=0 by A117,A118,FUNCT_1:72;
then A123: 1=(q3`1)^2 +0^2 by A121,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A123,JGRAPH_3:2;
case q3`1=1;
then A124: q3=|[1,0]| by A122,EUCLID:57
.=E-max(P) by A1,Th33;
0 in dom f2 by A95,RCOMP_1:15;
hence contradiction
by A83,A85,A92,A117,A120,A121,A124,FUNCT_1:def 8;
case q3`1=-1;
then A125: q3=|[-1,0]| by A122,EUCLID:57
.=W-min(P) by A1,Th32;
1 in dom f2 by A95,RCOMP_1:15;
hence contradiction
by A83,A85,A96,A117,A120,A121,A125,FUNCT_1:def 8;
end;
hence contradiction;
case t=r;
hence contradiction by A92,A93,A97,A99;
end;
hence contradiction;
end;
hence for q being Point of TOP-REAL 2
st q in Lower_Arc(P) holds q`2>=0;
end;
A126: Upper_Arc(P)
c= {p where p is Point of TOP-REAL 2:p in P & p`2>=0}
proof let x2 be set;assume
A127: x2 in Upper_Arc(P);
then reconsider q3=x2 as Point of TOP-REAL 2;
q3`2>=0 by A6,A7,A8,A23,A30,A37,A38,A39,A40,A48,A127;
hence x2 in {p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A32,A127;
end;
{p where p is Point of TOP-REAL 2:p in P & p`2>=0} c= Upper_Arc(P)
proof let x be set;assume
x in {p where p is Point of TOP-REAL 2:p in P & p`2>=0};
then consider p being Point of TOP-REAL 2 such that
A128: p=x & p in P & p`2>=0;
now per cases by A128;
case A129: p`2=0;
consider p8 being Point of TOP-REAL 2 such that
A130: p8=p & |.p8.|=1 by A1,A128;
A131: p=|[p`1,p`2]| by EUCLID:57;
1=sqrt((p`1)^2+(p`2)^2) by A130,JGRAPH_3:10
.=abs(p`1) by A129,SQUARE_1:60,91;
then (p`1)^2=1 by SQUARE_1:59,62;
then p=|[1,0]| or p=|[-1,0]| by A129,A131,JGRAPH_3:2;
hence x in Upper_Arc(P) by A1,A34,A35,A128,Th32,Th33;
case A132: p`2>0;
now assume not x in Upper_Arc(P);
then A133: x in Lower_Arc(P) by A6,A128,XBOOLE_0:def 2;
rng f2 =[#]((TOP-REAL 2)|R) by A83,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x2 being set such that
A134: x2 in dom f2 & p=f2.x2 by A128,A133,FUNCT_1:def 5;
dom f2= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x2 in {r where r is Real: 0<=r & r<=1 } by A134,RCOMP_1:def 1;
then consider t2 being Real such that
A135: x2=t2 & 0<=t2 & t2<=1;
A136: now assume t2=0;
then p=|[1,0]| by A1,A83,A134,A135,Th33;
hence contradiction by A132,EUCLID:56;
end;
now assume t2=1;
then p=|[-1,0]| by A1,A83,A134,A135,Th32;
hence contradiction by A132,EUCLID:56;
end;
then A137: 0<t2 & t2<1 & f2.t2=p & p`2>0 by A132,A134,A135,A136,REAL_1
:def 5;
A138: (|[0,-1]|)`1=0 by EUCLID:56;
A139: (|[0,-1]|)`2=-1 by EUCLID:56;
then |.|[0,-1]|.|=sqrt((0)^2+(-1)^2) by A138,JGRAPH_3:10
.=1 by SQUARE_1:59,60,61,83;
then A140: |[0,-1]| in {q where q is Point of TOP-REAL 2: |.q.|=1};
now per cases by A1,A6,A140,XBOOLE_0:def 2;
case |[0,-1]| in Upper_Arc(P);
hence contradiction by A6,A7,A8,A23,A30,A37,A38,A39,A40,A48,A139;
case |[0,-1]| in Lower_Arc(P);
hence contradiction by A91,A137,A139;
end;
hence contradiction;
end;
hence x in Upper_Arc(P);
end;
hence x in Upper_Arc(P);
end;
hence thesis by A126,XBOOLE_0:def 10;
end;
theorem Th38: for P being compact non empty Subset of TOP-REAL 2 st
P={q where q is Point of TOP-REAL 2: |.q.|=1} holds
Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={q where q is Point of TOP-REAL 2: |.q.|=1};
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
A4: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A2,JORDAN6:def 9;
set P4=Lower_Arc(P);
A5: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
A6: W-bound(P)=-1 by A1,Th31;
A7: E-bound(P)=1 by A1,Th31;
A8: Vertical_Line(0)={p where p is Point of TOP-REAL 2: p`1=0}
by JORDAN6:def 6;
reconsider P1=Lower_Arc(P) as Subset of TOP-REAL 2;
reconsider P2=Upper_Arc(P) as Subset of TOP-REAL 2;
reconsider Q=Vertical_Line(0) as Subset of TOP-REAL 2;
set p11=W-min(P);
set p22=E-max(P);
set pj= First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line(0));
set p8= Last_Point(Lower_Arc(P),E-max(P),W-min(P),
Vertical_Line(0));
A9: W-bound P=-1 by A1,Th31;
A10: E-bound P=1 by A1,Th31;
A11: S-bound P=-1 by A1,Th31;
A12: N-bound P=1 by A1,Th31;
then A13: LSeg(|[0,-1]|,|[0,1]|) meets Lower_Arc P
by A2,A9,A10,A11,JORDAN1B:27;
A14: LSeg(|[0,-1]|,|[0,1]|) c= Q
proof let x be set;assume x in LSeg(|[0,-1]|,|[0,1]|);
then x in {(1-l)*(|[0,-1]|) +l*(|[0,1]|) where l is Real: 0<=l & l
<=1}
by TOPREAL1:def 4;
then consider l being Real such that
A15: x=(1-l)*(|[0,-1]|) +l*(|[0,1]|) & 0<=l & l<=1;
((1-l)*(|[0,-1]|) +l*(|[0,1]|))`1
= ((1-l)*(|[0,-1]|))`1 +(l*(|[0,1]|))`1 by TOPREAL3:7
.=(1-l)*(|[0,-1]|)`1+(l*(|[0,1]|))`1 by TOPREAL3:9
.=(1-l)*(|[0,-1]|)`1+l*((|[0,1]|))`1 by TOPREAL3:9
.=(1-l)*0+l*((|[0,1]|))`1 by EUCLID:56
.=(1-l)*0+l*0 by EUCLID:56
.=0;
hence x in Q by A8,A15;
end;
then A16: P1 meets Q by A13,XBOOLE_1:64;
A17: Lower_Arc(P) is closed by A4,JORDAN6:12;
Vertical_Line(0) is closed by JORDAN6:33;
then P1 /\ Q is closed by A17,TOPS_1:35;
then A18: p8 in P1 /\ Q &
for g being map of I[01], (TOP-REAL 2)|P1, s2 being Real st
g is_homeomorphism & g.0 = p22 & g.1 = p11
& g.s2 = p8 & 0 <= s2 & s2 <= 1 holds
(for t being Real st 1 >= t & t > s2 holds not g.t in Q)
by A4,A16,JORDAN5C:def 2;
P1 /\ Q c= {|[0,-1]|,|[0,1]|}
proof let x be set;assume x in P1 /\ Q;
then A19: x in P1 & x in Q by XBOOLE_0:def 3;
then consider p being Point of TOP-REAL 2 such that
A20: p=x & p`1=0 by A8;
x in P by A5,A19,XBOOLE_0:def 2;
then consider q being Point of TOP-REAL 2 such that
A21: q=x & |.q.|=1 by A1;
0+(q`2)^2 =1 by A20,A21,JGRAPH_3:10,SQUARE_1:59,60;
then q`2=1 or q`2=-1 by JGRAPH_3:2;
then x=|[0,-1]| or x=|[0,1]| by A20,A21,EUCLID:57;
hence x in {|[0,-1]|,|[0,1]|} by TARSKI:def 2;
end;
then p8=|[0,-1]| or p8=|[0,1]| by A18,TARSKI:def 2;
then A22: p8`2=-1 or p8`2=1 by EUCLID:56;
LSeg(|[0,-1]|,|[0,1]|) meets Upper_Arc P
by A2,A9,A10,A11,A12,JORDAN1B:26;
then A23: P2 meets Q by A14,XBOOLE_1:64;
A24: Upper_Arc(P) is closed by A3,JORDAN6:12;
Vertical_Line(0) is closed by JORDAN6:33;
then P2 /\ Q is closed by A24,TOPS_1:35;
then A25: pj in P2 /\ Q &
for g being map of I[01], (TOP-REAL 2)|P2, s2 being Real st
g is_homeomorphism & g.0 = p11 & g.1 = p22
& g.s2 = pj & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds not g.t in Q
by A3,A23,JORDAN5C:def 1;
P2 /\ Q c= {|[0,-1]|,|[0,1]|}
proof let x be set;assume x in P2 /\ Q;
then A26: x in P2 & x in Q by XBOOLE_0:def 3;
then consider p being Point of TOP-REAL 2 such that
A27: p=x & p`1=0 by A8;
x in P by A5,A26,XBOOLE_0:def 2;
then consider q being Point of TOP-REAL 2 such that
A28: q=x & |.q.|=1 by A1;
0+(q`2)^2 =1 by A27,A28,JGRAPH_3:10,SQUARE_1:59,60;
then q`2=1 or q`2=-1 by JGRAPH_3:2;
then x=|[0,-1]| or x=|[0,1]| by A27,A28,EUCLID:57;
hence x in {|[0,-1]|,|[0,1]|} by TARSKI:def 2;
end;
then pj=|[0,-1]| or pj=|[0,1]| by A25,TARSKI:def 2;
then A29: pj`2=-1 or pj`2=1 by EUCLID:56;
A30: p8 in P1 by A18,XBOOLE_0:def 3;
A31: Lower_Arc(P) c= P by A5,XBOOLE_1:7;
A32: Upper_Arc(P) c= P by A5,XBOOLE_1:7;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A33: E-max(P) in Lower_Arc(P) by A5,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A34: W-min(P) in Lower_Arc(P) by A5,XBOOLE_0:def 3;
reconsider R=Lower_Arc(P)
as non empty Subset of TOP-REAL 2;
consider f being map of I[01], (TOP-REAL 2)|R such that
A35: f is_homeomorphism & f.0 =E-max(P) &
f.1 =W-min(P) by A4,TOPREAL1:def 2;
rng f =[#]((TOP-REAL 2)|R) by A35,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x8 being set such that
A36: x8 in dom f & p8=f.x8 by A30,FUNCT_1:def 5;
dom f= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x8 in {r where r is Real: 0<=r & r<=1 } by A36,RCOMP_1:def 1;
then consider r8 being Real such that
A37: x8=r8 & 0<=r8 & r8<=1;
A38: now assume r8=0;
then p8=|[1,0]| by A1,A35,A36,A37,Th33;
hence contradiction by A22,EUCLID:56;
end;
now assume r8=1;
then p8=|[-1,0]| by A1,A35,A36,A37,Th32;
hence contradiction by A22,EUCLID:56;
end;
then A39: 1>r8 by A37,REAL_1:def 5;
reconsider h2=proj2 as map of TOP-REAL 2,R^1 by JGRAPH_2:17;
A40: f is continuous by A35,TOPS_2:def 5;
A41: f is one-to-one by A35,TOPS_2:def 5;
for p being Point of (TOP-REAL 2) holds
h2.p=proj2.p;
then A42: h2 is continuous by Th35;
A43: dom f=the carrier of I[01] by FUNCT_2:def 1;
A44: the carrier of ((TOP-REAL 2)|R)=R by JORDAN1:1;
then A45: rng f c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
dom h2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then A46: dom (h2*f)=the carrier of I[01] by A43,A45,RELAT_1:46;
rng (h2*f) c= rng h2 by RELAT_1:45;
then rng (h2*f) c= the carrier of R^1 by XBOOLE_1:1;
then h2*f is Function of the carrier of I[01],the carrier of R^1
by A46,FUNCT_2:4;
then reconsider g0=h2*f as map of I[01],R^1;
A47: (ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f.t=p & p`2<0) implies for q being Point of TOP-REAL 2
st q in Lower_Arc(P) holds q`2<=0
proof assume
(ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f.t=p & p`2<0);
then consider p being Point of TOP-REAL 2,t being Real such that
A48: 0<t & t<1 & f.t=p & p`2<0;
now assume ex q being Point of TOP-REAL 2
st q in Lower_Arc(P) & q`2>0;
then consider q being Point of TOP-REAL 2 such that
A49: q in Lower_Arc(P) & q`2>0;
rng f =[#]((TOP-REAL 2)|R) by A35,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x being set such that
A50: x in dom f & q=f.x by A49,FUNCT_1:def 5;
A51: dom f= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x in {r where r is Real: 0<=r & r<=1 } by A50,RCOMP_1:def 1;
then consider r being Real such that
A52: x=r & 0<=r & r<=1;
A53: (h2*f).r=h2.q by A50,A52,FUNCT_1:23
.=q`2 by PSCOMP_1:def 29;
t in {v where v is Real: 0<=v & v<=1 } by A48;
then A54: t in [.0,1.] by RCOMP_1:def 1;
then A55: (h2*f).t=h2.p by A48,A51,FUNCT_1:23
.=p`2 by PSCOMP_1:def 29;
now per cases by REAL_1:def 5;
case A56: r<t;
[.r,t.] c= [.0,1.] by A50,A51,A52,A54,RCOMP_1:16;
then reconsider B=[.r,t.] as non empty Subset of I[01]
by A56,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g0 is continuous by A40,A42,Th10;
then A57: g is continuous by TOPMETR:10;
A58: Closed-Interval-TSpace(r,t)=I[01]|B by A48,A52,A56,Th6,TOPMETR:27;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A56;
then A59: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A56;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A48,A49,A53,A55,A59,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A60: g.r1=0 & r<r1 & r1<t by A56,A57,A58,TOPREAL5:14;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A60;
then A61: r1 in B by RCOMP_1:def 1;
A62: 0<r1 by A52,A60;
r1<1 by A48,A60,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A62;
then A63: r1 in dom f by A51,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 5;
then f.r1 in R by A44;
then f.r1 in P by A31;
then consider q3 being Point of TOP-REAL 2 such that
A64: q3=f.r1 & |.q3.|=1 by A1;
A65: q3`2=h2.(f.r1) by A64,PSCOMP_1:def 29
.=(h2*f).r1 by A63,FUNCT_1:23
.=0 by A60,A61,FUNCT_1:72;
then A66: 1=(q3`1)^2 +0^2 by A64,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A66,JGRAPH_3:2;
case q3`1=1;
then A67: q3=|[1,0]| by A65,EUCLID:57
.=E-max(P) by A1,Th33;
0 in dom f by A51,RCOMP_1:15;
hence contradiction
by A35,A41,A52,A60,A63,A64,A67,FUNCT_1:def 8;
case q3`1=-1;
then A68: q3=|[-1,0]| by A65,EUCLID:57
.=W-min(P) by A1,Th32;
1 in dom f by A51,RCOMP_1:15;
hence contradiction
by A35,A41,A48,A60,A63,A64,A68,FUNCT_1:def 8;
end;
hence contradiction;
case A69: t<r;
[.t,r.] c= [.0,1.] by A50,A51,A52,A54,RCOMP_1:16;
then reconsider B=[.t,r.] as non empty Subset of I[01]
by A69,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g0 is continuous by A40,A42,Th10;
then A70: g is continuous by TOPMETR:10;
A71: Closed-Interval-TSpace(t,r)=I[01]|B by A48,A52,A69,Th6,TOPMETR:27;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A69;
then A72: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A69;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A48,A49,A53,A55,A72,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A73: g.r1=0 & t<r1 & r1<r by A69,A70,A71,TOPREAL5:14;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A73;
then A74: r1 in B by RCOMP_1:def 1;
A75: 0<r1 by A48,A73,AXIOMS:22;
r1<1 by A52,A73,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A75;
then A76: r1 in dom f by A51,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 5;
then f.r1 in R by A44;
then f.r1 in P by A31;
then consider q3 being Point of TOP-REAL 2 such that
A77: q3=f.r1 & |.q3.|=1 by A1;
A78: q3`2=h2.(f.r1) by A77,PSCOMP_1:def 29
.=(h2*f).r1 by A76,FUNCT_1:23
.=0 by A73,A74,FUNCT_1:72;
then A79: 1=(q3`1)^2 +0^2 by A77,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A79,JGRAPH_3:2;
case q3`1=1;
then A80: q3=|[1,0]| by A78,EUCLID:57
.=E-max(P) by A1,Th33;
0 in dom f by A51,RCOMP_1:15;
hence contradiction
by A35,A41,A48,A73,A76,A77,A80,FUNCT_1:def 8;
case q3`1=-1;
then A81: q3=|[-1,0]| by A78,EUCLID:57
.=W-min(P) by A1,Th32;
1 in dom f by A51,RCOMP_1:15;
hence contradiction
by A35,A41,A52,A73,A76,A77,A81,FUNCT_1:def 8;
end;
hence contradiction;
case t=r;
hence contradiction by A48,A49,A53,A55;
end;
hence contradiction;
end;
hence for q being Point of TOP-REAL 2
st q in Lower_Arc(P) holds q`2<=0;
end;
reconsider R=Upper_Arc(P) as non empty Subset of TOP-REAL 2;
consider f2 being map of I[01], (TOP-REAL 2)|R such that
A82: f2 is_homeomorphism & f2.0 =W-min(P) &
f2.1 =E-max(P) by A3,TOPREAL1:def 2;
A83: f2 is continuous by A82,TOPS_2:def 5;
A84: f2 is one-to-one by A82,TOPS_2:def 5;
for p being Point of (TOP-REAL 2) holds h2.p=proj2.p;
then A85: h2 is continuous by Th35;
A86: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A87: the carrier of ((TOP-REAL 2)|R)=R by JORDAN1:1;
then A88: rng f2 c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
dom h2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then A89: dom (h2*f2)=the carrier of I[01] by A86,A88,RELAT_1:46;
rng (h2*f2) c= rng h2 by RELAT_1:45;
then rng (h2*f2) c= the carrier of R^1 by XBOOLE_1:1;
then h2*f2 is Function of the carrier of I[01],the carrier of R^1
by A89,FUNCT_2:4;
then reconsider g1=h2*f2 as map of I[01],R^1;
A90: (ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f2.t=p & p`2<0) implies for q being Point of TOP-REAL 2
st q in Upper_Arc(P) holds q`2<=0
proof assume
(ex p being Point of TOP-REAL 2,t being Real st
0<t & t<1 & f2.t=p & p`2<0);
then consider p being Point of TOP-REAL 2,t being Real such that
A91: 0<t & t<1 & f2.t=p & p`2<0;
now assume ex q being Point of TOP-REAL 2
st q in Upper_Arc(P) & q`2>0;
then consider q being Point of TOP-REAL 2 such that
A92: q in Upper_Arc(P) & q`2>0;
rng f2 =[#]((TOP-REAL 2)|R) by A82,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x being set such that
A93: x in dom f2 & q=f2.x by A92,FUNCT_1:def 5;
A94: dom f2= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x in {r where r is Real: 0<=r & r<=1 } by A93,RCOMP_1:def 1;
then consider r being Real such that
A95: x=r & 0<=r & r<=1;
A96: (h2*f2).r=h2.q by A93,A95,FUNCT_1:23
.=q`2 by PSCOMP_1:def 29;
t in {v where v is Real: 0<=v & v<=1 } by A91;
then A97: t in [.0,1.] by RCOMP_1:def 1;
then A98: (h2*f2).t=h2.p by A91,A94,FUNCT_1:23
.=p`2 by PSCOMP_1:def 29;
now per cases by REAL_1:def 5;
case A99: r<t;
[.r,t.] c= [.0,1.] by A93,A94,A95,A97,RCOMP_1:16;
then reconsider B=[.r,t.] as non empty Subset of I[01]
by A99,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g1 is continuous by A83,A85,Th10;
then A100: g is continuous by TOPMETR:10;
A101: Closed-Interval-TSpace(r,t)=I[01]|B by A91,A95,A99,Th6,TOPMETR:27;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A99;
then A102: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A99;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A91,A92,A96,A98,A102,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A103: g.r1=0 & r<r1 & r1<t by A99,A100,A101,TOPREAL5:14;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A103;
then A104: r1 in B by RCOMP_1:def 1;
A105: 0<r1 by A95,A103;
r1<1 by A91,A103,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A105;
then A106: r1 in dom f2 by A94,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 5;
then f2.r1 in R by A87;
then f2.r1 in P by A32;
then consider q3 being Point of TOP-REAL 2 such that
A107: q3=f2.r1 & |.q3.|=1 by A1;
A108: q3`2=h2.(f2.r1) by A107,PSCOMP_1:def 29
.=(h2*f2).r1 by A106,FUNCT_1:23
.=0 by A103,A104,FUNCT_1:72;
then A109: 1=(q3`1)^2 +0^2 by A107,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A109,JGRAPH_3:2;
case q3`1=1;
then A110: q3=|[1,0]| by A108,EUCLID:57
.=E-max(P) by A1,Th33;
1 in dom f2 by A94,RCOMP_1:15;
hence contradiction
by A82,A84,A91,A103,A106,A107,A110,FUNCT_1:def 8;
case q3`1=-1;
then A111: q3=|[-1,0]| by A108,EUCLID:57
.=W-min(P) by A1,Th32;
0 in dom f2 by A94,RCOMP_1:15;
hence contradiction
by A82,A84,A95,A103,A106,A107,A111,FUNCT_1:def 8;
end;
hence contradiction;
case A112: t<r;
[.t,r.] c= [.0,1.] by A93,A94,A95,A97,RCOMP_1:16;
then reconsider B=[.t,r.] as non empty Subset of I[01]
by A112,BORSUK_1:83,RCOMP_1:15;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as map of (I[01]|B0),R^1 by JGRAPH_3:12;
g1 is continuous by A83,A85,Th10;
then A113: g is continuous by TOPMETR:10;
A114: Closed-Interval-TSpace(t,r)=I[01]|B by A91,A95,A112,Th6,TOPMETR:27
;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A112;
then A115: r in B by RCOMP_1:def 1;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A112;
then t in B by RCOMP_1:def 1;
then (q`2)*(p`2)<0 & q`2=g.r & p`2=g.t by A91,A92,A96,A98,A115,FUNCT_1:
72,SQUARE_1:24;
then consider r1 being Real such that
A116: g.r1=0 & t<r1 & r1<r by A112,A113,A114,TOPREAL5:14;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A116;
then A117: r1 in B by RCOMP_1:def 1;
A118: 0<r1 by A91,A116,AXIOMS:22;
r1<1 by A95,A116,AXIOMS:22;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A118;
then A119: r1 in dom f2 by A94,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 5;
then f2.r1 in R by A87;
then f2.r1 in P by A32;
then consider q3 being Point of TOP-REAL 2 such that
A120: q3=f2.r1 & |.q3.|=1 by A1;
A121: q3`2=h2.(f2.r1) by A120,PSCOMP_1:def 29
.=(h2*f2).r1 by A119,FUNCT_1:23
.=0 by A116,A117,FUNCT_1:72;
then A122: 1=(q3`1)^2 +0^2 by A120,JGRAPH_3:10,SQUARE_1:59
.=(q3`1)^2 by SQUARE_1:60;
now per cases by A122,JGRAPH_3:2;
case q3`1=1;
then A123: q3=|[1,0]| by A121,EUCLID:57
.=E-max(P) by A1,Th33;
1 in dom f2 by A94,RCOMP_1:15;
hence contradiction
by A82,A84,A95,A116,A119,A120,A123,FUNCT_1:def 8;
case q3`1=-1;
then A124: q3=|[-1,0]| by A121,EUCLID:57
.=W-min(P) by A1,Th32;
0 in dom f2 by A94,RCOMP_1:15;
hence contradiction
by A82,A84,A91,A116,A119,A120,A124,FUNCT_1:def 8;
end;
hence contradiction;
case t=r;
hence contradiction by A91,A92,A96,A98;
end;
hence contradiction;
end;
hence for q being Point of TOP-REAL 2
st q in Upper_Arc(P) holds q`2<=0;
end;
A125: Lower_Arc(P)
c= {p where p is Point of TOP-REAL 2:p in P & p`2<=0}
proof let x2 be set;assume
A126: x2 in Lower_Arc(P);
then reconsider q3=x2 as Point of TOP-REAL 2;
q3`2<=0 by A5,A6,A7,A22,A29,A36,A37,A38,A39,A47,A126;
hence x2 in {p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A31,A126;
end;
{p where p is Point of TOP-REAL 2:p in P & p`2<=0} c= Lower_Arc(P)
proof let x be set;assume
x in {p where p is Point of TOP-REAL 2:p in P & p`2<=0};
then consider p being Point of TOP-REAL 2 such that
A127: p=x & p in P & p`2<=0;
now per cases by A127;
case A128: p`2=0;
consider p8 being Point of TOP-REAL 2 such that
A129: p8=p & |.p8.|=1 by A1,A127;
A130: p=|[p`1,p`2]| by EUCLID:57;
1=sqrt((p`1)^2+(p`2)^2) by A129,JGRAPH_3:10
.=abs(p`1) by A128,SQUARE_1:60,91;
then (p`1)^2=1 by SQUARE_1:59,62;
then p=|[1,0]| or p=|[-1,0]| by A128,A130,JGRAPH_3:2;
hence x in Lower_Arc(P) by A1,A33,A34,A127,Th32,Th33;
case A131: p`2<0;
now assume not x in Lower_Arc(P);
then A132: x in Upper_Arc(P) by A5,A127,XBOOLE_0:def 2;
rng f2 =[#]((TOP-REAL 2)|R) by A82,TOPS_2:def 5
.=R by PRE_TOPC:def 10;
then consider x2 being set such that
A133: x2 in dom f2 & p=f2.x2 by A127,A132,FUNCT_1:def 5;
dom f2= [.0,1.]
by BORSUK_1:83,FUNCT_2:def 1;
then x2 in {r where r is Real: 0<=r & r<=1 } by A133,RCOMP_1:def 1;
then consider t2 being Real such that
A134: x2=t2 & 0<=t2 & t2<=1;
A135: now assume t2=1;
then p=|[1,0]| by A1,A82,A133,A134,Th33;
hence contradiction by A131,EUCLID:56;
end;
now assume t2=0;
then p=|[-1,0]| by A1,A82,A133,A134,Th32;
hence contradiction by A131,EUCLID:56;
end;
then A136: 0<t2 & t2<1 & f2.t2=p & p`2<0 by A131,A133,A134,A135,REAL_1
:def 5;
A137: (|[0,1]|)`1=0 by EUCLID:56;
A138: (|[0,1]|)`2=1 by EUCLID:56;
then |.|[0,1]|.|=sqrt((0)^2+(1)^2) by A137,JGRAPH_3:10
.=1 by SQUARE_1:59,60,83;
then A139: |[0,1]| in {q where q is Point of TOP-REAL 2: |.q.|=1};
now per cases by A1,A5,A139,XBOOLE_0:def 2;
case |[0,1]| in Lower_Arc(P);
hence contradiction by A5,A6,A7,A22,A29,A36,A37,A38,A39,A47,A138;
case |[0,1]| in Upper_Arc(P);
hence contradiction by A90,A136,A138;
end;
hence contradiction;
end;
hence x in Lower_Arc(P);
end;
hence x in Lower_Arc(P);
end;
hence thesis by A125,XBOOLE_0:def 10;
end;
theorem Th39: for a,b,d,e being Real st a<=b & e>0
ex f being map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*a+d,e*b+d)
st f is_homeomorphism & for r being Real st r in [.a,b.] holds
f.r=e*r+d
proof let a,b,d,e be Real;assume A1: a<=b & e>0;
then e*a<=e*b by AXIOMS:25;
then A2: e*a+d<=e*b+d by REAL_1:55;
set X=the carrier of Closed-Interval-TSpace(a,b);
set Y=the carrier of Closed-Interval-TSpace(e*a+d,e*b+d);
defpred P[set,set] means (for r being Real st $1=r holds $2=e*r+d);
A3: X=[.a,b.] by A1,TOPMETR:25;
A4: Y=[.e*a+d,e*b+d.] by A2,TOPMETR:25;
A5: for x being set st x in X ex y being set st y in Y & P[x,y]
proof let x be set;assume A6: x in X;
then reconsider r1=x as Real by A3;
A7: a<=r1 & r1<=b by A3,A6,TOPREAL5:1;
then A8: e*a<=e*r1 by A1,AXIOMS:25;
A9: e*r1<=e*b by A1,A7,AXIOMS:25;
set y1=e*r1+d;
A10: for r being Real st x=r holds y1=e*r+d;
e*a+d<=y1 & y1 <=e*b+d by A8,A9,REAL_1:55;
then y1 in Y by A4,TOPREAL5:1;
hence ex y being set st y in Y & P[x,y] by A10;
end;
ex f being Function of X,Y st for x being set st x in X holds P[x,f.x]
from FuncEx1(A5);
then consider f1 being Function of X,Y such that
A11: for x being set st x in X holds P[x,f1.x];
reconsider f2=f1 as map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*a+d,e*b+d);
A12: for r being Real st r in [.a,b.] holds f2.r=e*r+d by A3,A11;
set S=Closed-Interval-TSpace(a,b);
set T=Closed-Interval-TSpace(e*a+d,e*b+d);
A13: dom f2=the carrier of S by FUNCT_2:def 1;
then A14: dom f2=[#]S by PRE_TOPC:12;
for x1,x2 being set st x1 in dom f2 & x2 in dom f2 & f2.x1=f2.x2
holds x1=x2
proof let x1,x2 be set;assume
A15: x1 in dom f2 & x2 in dom f2 & f2.x1=f2.x2;
then reconsider r1=x1 as Real by A3,A13;
reconsider r2=x2 as Real by A3,A13,A15;
f2.x1=e*r1+d by A11,A13,A15;
then e*r1+d-d=e*r2+d-d by A11,A13,A15 .=e*r2 by XCMPLX_1:26;
then e*r1=e*r2 by XCMPLX_1:26;
then r1*e/e=r2 by A1,XCMPLX_1:90;
hence x1=x2 by A1,XCMPLX_1:90;
end;
then A16: f2 is one-to-one by FUNCT_1:def 8;
rng f2 c= the carrier of T;
then A17: rng f2 c= [#]T by PRE_TOPC:12;
[#]T c= rng f2
proof let y be set;assume A18: y in [#]T;
then y in [.e*a+d,e*b+d.] by A4;
then reconsider ry=y as Real;
A19: e*a+d <= ry & ry<=e*b+d by A4,A18,TOPREAL5:1;
then e*a+d-d<=ry-d by REAL_1:49;
then e*a<=ry-d by XCMPLX_1:26;
then a*e/e<=(ry-d)/e by A1,REAL_1:73;
then A20: a<=(ry-d)/e by A1,XCMPLX_1:90;
e*b+d-d>=ry-d by A19,REAL_1:49;
then e*b>=ry-d by XCMPLX_1:26;
then b*e/e>=(ry-d)/e by A1,REAL_1:73;
then b>=(ry-d)/e by A1,XCMPLX_1:90;
then A21: (ry-d)/e in [.a,b.] by A20,TOPREAL5:1;
then f2.((ry-d)/e)=e*((ry-d)/e)+d by A3,A11
.=ry-d+d by A1,XCMPLX_1:88.=ry by XCMPLX_1:27;
hence y in rng f2 by A3,A13,A21,FUNCT_1:12;
end;
then A22: rng f2 = [#]T by A17,XBOOLE_0:def 10;
then rng f2=Y by PRE_TOPC:12;
then f1 is Function of the carrier of S,the carrier of R^1
by A4,A13,FUNCT_2:4,TOPMETR:24;
then reconsider f3=f1 as map of S,R^1;
defpred P1[set,set] means for r being Real st r=$1 holds $2=e*r+d;
A23: for x being set st x in the carrier of R^1
ex y being set st y in the carrier of R^1 & P1[x,y]
proof let x be set;assume x in the carrier of R^1;
then reconsider rx=x as Real by TOPMETR:24;
reconsider ry=e*rx+d as Real;
for r being Real st r=x holds ry=e*r+d;
hence ex y being set st y in the carrier of R^1 & P1[x,y] by TOPMETR:24
;
end;
ex f4 being Function of the carrier of R^1,the carrier of R^1 st
for x being set st x in the carrier of R^1 holds P1[x,f4.x]
from FuncEx1(A23);
then consider f4 being Function of the carrier of R^1,the carrier of R^1
such that
A24: for x being set st x in the carrier of R^1 holds P1[x,f4.x];
reconsider f5=f4 as map of R^1,R^1;
for x being Real holds f5.x = e*x + d by A24,TOPMETR:24;
then A25: f5 is continuous by TOPMETR:28;
reconsider B=the carrier of S as Subset of R^1
by A3,TOPMETR:24;
A26: R^1|B= S by A1,A3,TOPMETR:26;
A27: dom f3=B by FUNCT_2:def 1;
A28: (dom f5) /\ B
=REAL /\ B by FUNCT_2:def 1,TOPMETR:24
.=B by TOPMETR:24,XBOOLE_1:28;
for x being set st x in dom f3 holds f3.x=f5.x
proof let x be set;assume
A29: x in dom f3;
then A30: x in the carrier of S by FUNCT_2:def 1;
reconsider rx=x as Real by A3,A13,A29;
f4.x=e*rx+d by A24,TOPMETR:24;
hence f3.x=f5.x by A11,A30;
end;
then f3=f5|B by A27,A28,FUNCT_1:68;
then A31: f3 is continuous by A25,A26,TOPMETR:10;
reconsider C=the carrier of T as Subset of R^1
by A4,TOPMETR:24;
R^1|C=T by A2,A4,TOPMETR:26;
then A32: f2 is continuous by A31,TOPMETR:9;
A33: S is compact by A1,HEINE:11;
T=TopSpaceMetr(Closed-Interval-MSpace(e*a+d,e*b+d)) by TOPMETR:def 8;
then T is_T2 by PCOMPS_1:38;
then f2 is_homeomorphism by A14,A16,A22,A32,A33,COMPTS_1:26;
hence thesis by A12;
end;
theorem Th40: for a,b,d,e being Real st a<=b & e<0
ex f being map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*b+d,e*a+d)
st f is_homeomorphism & for r being Real st r in [.a,b.] holds
f.r=e*r+d
proof let a,b,d,e be Real;assume A1: a<=b & e<0;
then e*a>=e*b by REAL_1:52;
then A2: e*a+d>=e*b+d by REAL_1:55;
set X=the carrier of Closed-Interval-TSpace(a,b);
set Y=the carrier of Closed-Interval-TSpace(e*b+d,e*a+d);
defpred P[set,set] means (for r being Real st $1=r holds $2=e*r+d);
A3: X=[.a,b.] by A1,TOPMETR:25;
A4: Y=[.e*b+d,e*a+d.] by A2,TOPMETR:25;
A5: for x being set st x in X ex y being set st y in Y & P[x,y]
proof let x be set;assume A6: x in X;
then reconsider r1=x as Real by A3;
A7: a<=r1 & r1<=b by A3,A6,TOPREAL5:1;
then A8: e*a>=e*r1 by A1,REAL_1:52;
A9: e*r1>=e*b by A1,A7,REAL_1:52;
set y1=e*r1+d;
A10: for r being Real st x=r holds y1=e*r+d;
e*a+d>=y1 & y1 >=e*b+d by A8,A9,REAL_1:55;
then y1 in Y by A4,TOPREAL5:1;
hence ex y being set st y in Y & P[x,y] by A10;
end;
ex f being Function of X,Y st for x being set st x in X holds P[x,f.x]
from FuncEx1(A5);
then consider f1 being Function of X,Y such that
A11: for x being set st x in X holds P[x,f1.x];
reconsider f2=f1 as map of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*b+d,e*a+d);
A12: for r being Real st r in [.a,b.] holds f2.r=e*r+d by A3,A11;
set S=Closed-Interval-TSpace(a,b);
set T=Closed-Interval-TSpace(e*b+d,e*a+d);
A13: dom f2=the carrier of S by FUNCT_2:def 1;
then A14: dom f2=[#]S by PRE_TOPC:12;
for x1,x2 being set st x1 in dom f2 & x2 in dom f2 & f2.x1=f2.x2
holds x1=x2
proof let x1,x2 be set;assume
A15: x1 in dom f2 & x2 in dom f2 & f2.x1=f2.x2;
then reconsider r1=x1 as Real by A3,A13;
reconsider r2=x2 as Real by A3,A13,A15;
f2.x1=e*r1+d by A11,A13,A15;
then e*r1+d-d=e*r2+d-d by A11,A13,A15 .=e*r2 by XCMPLX_1:26;
then e*r1=e*r2 by XCMPLX_1:26;
then r1*e/e=r2 by A1,XCMPLX_1:90;
hence x1=x2 by A1,XCMPLX_1:90;
end;
then A16: f2 is one-to-one by FUNCT_1:def 8;
rng f2 c= the carrier of T;
then A17: rng f2 c= [#]T by PRE_TOPC:12;
[#]T c= rng f2
proof let y be set;assume A18: y in [#]T;
then y in [.e*b+d,e*a+d.] by A4;
then reconsider ry=y as Real;
A19: e*b+d <= ry & ry<=e*a+d by A4,A18,TOPREAL5:1;
then e*a+d-d>=ry-d by REAL_1:49;
then e*a>=ry-d by XCMPLX_1:26;
then a*e/e<=(ry-d)/e by A1,REAL_1:74;
then A20: a<=(ry-d)/e by A1,XCMPLX_1:90;
e*b+d-d<=ry-d by A19,REAL_1:49;
then e*b<=ry-d by XCMPLX_1:26;
then b*e/e>=(ry-d)/e by A1,REAL_1:74;
then b>=(ry-d)/e by A1,XCMPLX_1:90;
then A21: (ry-d)/e in [.a,b.] by A20,TOPREAL5:1;
then f2.((ry-d)/e)=e*((ry-d)/e)+d by A3,A11
.=ry-d+d by A1,XCMPLX_1:88.=ry by XCMPLX_1:27;
hence y in rng f2 by A3,A13,A21,FUNCT_1:12;
end;
then A22: rng f2 = [#]T by A17,XBOOLE_0:def 10;
then rng f2=Y by PRE_TOPC:12;
then f1 is Function of the carrier of S,the carrier of R^1
by A4,A13,FUNCT_2:4,TOPMETR:24;
then reconsider f3=f1 as map of S,R^1;
defpred P1[set,set] means for r being Real st r=$1 holds $2=e*r+d;
A23: for x being set st x in the carrier of R^1
ex y being set st y in the carrier of R^1 & P1[x,y]
proof let x be set;assume x in the carrier of R^1;
then reconsider rx=x as Real by TOPMETR:24;
reconsider ry=e*rx+d as Real;
for r being Real st r=x holds ry=e*r+d;
hence ex y being set st y in the carrier of R^1 & P1[x,y]
by TOPMETR:24;
end;
ex f4 being Function of the carrier of R^1,the carrier of R^1 st
for x being set st x in the carrier of R^1 holds P1[x,f4.x]
from FuncEx1(A23);
then consider f4 being Function of the carrier of R^1,the carrier of R^1
such that
A24: for x being set st x in the carrier of R^1 holds P1[x,f4.x];
reconsider f5=f4 as map of R^1,R^1;
for x being Real holds f5.x = e*x + d by A24,TOPMETR:24;
then A25: f5 is continuous by TOPMETR:28;
reconsider B=the carrier of S as Subset of R^1
by A3,TOPMETR:24;
A26: R^1|B= S by A1,A3,TOPMETR:26;
A27: dom f3=B by FUNCT_2:def 1;
A28:(dom f5) /\ B =REAL /\ B by FUNCT_2:def 1,TOPMETR:24
.=B by TOPMETR:24,XBOOLE_1:28;
for x being set st x in dom f3 holds f3.x=f5.x
proof let x be set;assume
A29: x in dom f3;
then A30: x in the carrier of S by FUNCT_2:def 1;
reconsider rx=x as Real by A3,A13,A29;
f4.x=e*rx+d by A24,TOPMETR:24;
hence f3.x=f5.x by A11,A30;
end;
then f3=f5|B by A27,A28,FUNCT_1:68;
then A31: f3 is continuous by A25,A26,TOPMETR:10;
reconsider C=the carrier of T as Subset of R^1
by A4,TOPMETR:24;
R^1|C=T by A2,A4,TOPMETR:26;
then A32: f2 is continuous by A31,TOPMETR:9;
A33: S is compact by A1,HEINE:11;
T=TopSpaceMetr(Closed-Interval-MSpace(e*b+d,e*a+d))
by TOPMETR:def 8;
then T is_T2 by PCOMPS_1:38;
then f2 is_homeomorphism by A14,A16,A22,A32,A33,COMPTS_1:26;
hence thesis by A12;
end;
theorem Th41:
ex f being map of I[01],Closed-Interval-TSpace(-1,1)
st f is_homeomorphism & (for r being Real st r in [.0,1.] holds
f.r=(-2)*r+1) & f.0=1 & f.1=-1
proof
consider f being map of I[01],
Closed-Interval-TSpace((-2)*1+1,(-2)*0+1) such that
A1: f is_homeomorphism & (for r being Real st r in [.0,1.] holds
f.r=(-2)*r+1) by Th40,TOPMETR:27;
A2: f.0=(-2)*0+1 by A1,Lm1;
1 in [.0,1.] by TOPREAL5:1;
then f.1=-1 by A1;
hence thesis by A1,A2;
end;
theorem Th42:
ex f being map of I[01],Closed-Interval-TSpace(-1,1)
st f is_homeomorphism & (for r being Real st r in [.0,1.] holds
f.r=2*r-1) & f.0=-1 & f.1=1
proof
consider f being map of I[01],
Closed-Interval-TSpace(2*0+-1,2*1+-1) such that
A1: f is_homeomorphism & (for r being Real st r in [.0,1.] holds
f.r=2*r+-1) by Th39,TOPMETR:27;
A2: for r being Real st r in [.0,1.] holds f.r=2*r-1
proof let r be Real;assume r in [.0,1.];
hence f.r=2*r+-1 by A1 .=2*r-1 by XCMPLX_0:def 8;
end;
then A3: f.0=2*0-1 by Lm1 .=-1;
1 in [.0,1.] by TOPREAL5:1;
then f.1=2*1-1 by A2 .=1;
hence thesis by A1,A2,A3;
end;
Lm4:
now let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1};
reconsider g=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Lower_Arc(P);
reconsider g2=g|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
A2: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by JORDAN1:1;
hence g2.p=proj1.p by FUNCT_1:72;
end;
then A3: g2 is continuous by JGRAPH_2:39;
A4: dom g2=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A5: dom g2=K0 by JORDAN1:1;
A6: K0 c= P by A1,Th36;
A7: rng g2 c= the carrier of Closed-Interval-TSpace(-1,1)
proof let x be set;assume x in rng g2;
then consider z being set such that
A8: z in dom g2 & x=g2.z by FUNCT_1:def 5;
z in P by A5,A6,A8;
then consider p being Point of TOP-REAL 2 such that
A9: p=z & |.p.|=1 by A1;
A10: x=proj1.p by A2,A4,A8,A9 .=p`1 by PSCOMP_1:def 28;
1=(p`1)^2+(p`2)^2 by A9,JGRAPH_3:10,SQUARE_1:59;
then 1-(p`1)^2=(p`2)^2 by XCMPLX_1:26;
then 1-(p`1)^2>=0 by SQUARE_1:72;
then -(1-(p`1)^2)<=0 by REAL_1:66;
then (p`1)^2-1<=0 by XCMPLX_1:143;
then -1<=p`1 & p`1<=1 by JGRAPH_3:5;
then x in [.-1,1.] by A10,TOPREAL5:1;
hence x in the carrier of Closed-Interval-TSpace(-1,1) by TOPMETR:25;
end;
then g2 is Function of the carrier of ((TOP-REAL 2)|K0),
the carrier of Closed-Interval-TSpace(-1,1) by A4,FUNCT_2:4;
then reconsider g3=g2 as map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1);
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
then A11: dom g3=K0 by PRE_TOPC:def 10;
A12: rng g2 c= [#](Closed-Interval-TSpace(-1,1)) by A7,PRE_TOPC:12;
A13: [#](Closed-Interval-TSpace(-1,1)) c= rng g3
proof let x be set;assume x in [#](Closed-Interval-TSpace(-1,1));
then x in the carrier of (Closed-Interval-TSpace(-1,1));
then A14: x in [.-1,1.] by TOPMETR:25;
then reconsider r=x as Real;
set q=|[r,-sqrt(1-r^2)]|;
A15: |.q.|=sqrt((q`1)^2+(q`2)^2) by JGRAPH_3:10
.=sqrt(r^2+(q`2)^2) by EUCLID:56
.=sqrt(r^2+(-sqrt(1-r^2))^2) by EUCLID:56
.=sqrt(r^2+(sqrt(1-r^2))^2) by SQUARE_1:61;
-1<=r & r<=1 by A14,TOPREAL5:1;
then 1^2>=r^2 by JGRAPH_2:7;
then A16: 1-r^2>=0 by SQUARE_1:12,59;
then 0<=sqrt(1-r^2) by SQUARE_1:def 4;
then A17: -sqrt(1-r^2)<=0 by REAL_1:66;
|.q.|=sqrt(r^2+(1-r^2)) by A15,A16,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then A18: q in P by A1;
q`2=-sqrt(1-r^2) by EUCLID:56;
then q in {p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A17,A18
;
then A19: q in dom g3 by A1,A11,Th38;
then g3.q=proj1.q by A2,A4 .=q`1 by PSCOMP_1:def 28.=r by EUCLID:56;
hence x in rng g3 by A19,FUNCT_1:def 5;
end;
reconsider B=[.-1,1.] as non empty Subset of R^1
by TOPMETR:24,TOPREAL5:1;
A20: Closed-Interval-TSpace(-1,1)=R^1|B by TOPMETR:26;
for x,y being set st x in dom g3 & y in dom g3 & g3.x=g3.y holds x=y
proof let x,y be set;assume
A21: x in dom g3 & y in dom g3 & g3.x=g3.y;
then reconsider p1=x as Point of TOP-REAL 2 by A11;
reconsider p2=y as Point of TOP-REAL 2 by A11,A21;
A22: g3.x=proj1.p1 by A2,A4,A21 .=p1`1 by PSCOMP_1:def 28;
A23: g3.y=proj1.p2 by A2,A4,A21 .=p2`1 by PSCOMP_1:def 28;
A24: p1 in P by A6,A11,A21;
p2 in P by A6,A11,A21;
then consider p22 being Point of TOP-REAL 2 such that
A25: p2=p22 & |.p22.|=1 by A1;
1^2= (p2`1)^2+(p2`2)^2 by A25,JGRAPH_3:10;
then A26: 1^2-(p2`1)^2= (p2`2)^2 by XCMPLX_1:26;
consider p11 being Point of TOP-REAL 2 such that
A27: p1=p11 & |.p11.|=1 by A1,A24;
1^2= (p1`1)^2+(p1`2)^2 by A27,JGRAPH_3:10;
then 1^2-(p1`1)^2= (p1`2)^2 by XCMPLX_1:26;
then (-(p1`2))^2 =(p2`2)^2 by A21,A22,A23,A26,SQUARE_1:61;
then A28: (-(p1`2))^2 =(-(p2`2))^2 by SQUARE_1:61;
p1 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2<=0}
by A1,A11,A21,Th38;
then consider p33 being Point of TOP-REAL 2 such that
A29: p1=p33 & p33 in P & p33`2<=0;
--(p1`2)<=0 by A29;
then -(p1`2)>=0 by REAL_1:66;
then A30: -(p1`2)=sqrt((-(p2`2))^2) by A28,SQUARE_1:89;
p2 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2<=0}
by A1,A11,A21,Th38;
then consider p44 being Point of TOP-REAL 2 such that
A31: p2=p44 & p44 in P & p44`2<=0;
--(p2`2)<=0 by A31;
then -(p2`2)>=0 by REAL_1:66;
then -(p1`2)=-(p2`2) by A30,SQUARE_1:89;
then --(p1`2)=(p2`2);
then p1=|[p2`1,p2`2]| by A21,A22,A23,EUCLID:57
.=p2 by EUCLID:57;
hence x=y;
end;
hence proj1|K0 is continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) &
proj1|K0 is one-to-one &
rng (proj1|K0)=[#](Closed-Interval-TSpace(-1,1)) by A3,A12,A13,A20,FUNCT_1:
def 8,JGRAPH_1:63,XBOOLE_0:def 10;
end;
Lm5:
now let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1};
reconsider g=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Upper_Arc(P);
reconsider g2=g|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
A2: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by JORDAN1:1;
hence g2.p=proj1.p by FUNCT_1:72;
end;
then A3: g2 is continuous by JGRAPH_2:39;
A4: dom g2=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A5: dom g2=K0 by JORDAN1:1;
A6: K0 c= P by A1,Th36;
A7: rng g2 c= the carrier of Closed-Interval-TSpace(-1,1)
proof let x be set;assume x in rng g2;
then consider z being set such that
A8: z in dom g2 & x=g2.z by FUNCT_1:def 5;
z in P by A5,A6,A8;
then consider p being Point of TOP-REAL 2 such that
A9: p=z & |.p.|=1 by A1;
A10: x=proj1.p by A2,A4,A8,A9 .=p`1 by PSCOMP_1:def 28;
1=(p`1)^2+(p`2)^2 by A9,JGRAPH_3:10,SQUARE_1:59;
then 1-(p`1)^2=(p`2)^2 by XCMPLX_1:26;
then 1-(p`1)^2>=0 by SQUARE_1:72;
then -(1-(p`1)^2)<=0 by REAL_1:66;
then (p`1)^2-1<=0 by XCMPLX_1:143;
then -1<=p`1 & p`1<=1 by JGRAPH_3:5;
then x in [.-1,1.] by A10,TOPREAL5:1;
hence x in the carrier of Closed-Interval-TSpace(-1,1)
by TOPMETR:25;
end;
then g2 is Function of the carrier of ((TOP-REAL 2)|K0),
the carrier of Closed-Interval-TSpace(-1,1) by A4,FUNCT_2:4;
then reconsider g3=g2 as map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1);
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
then A11: dom g3=K0 by PRE_TOPC:def 10;
A12: rng g2 c= [#](Closed-Interval-TSpace(-1,1)) by A7,PRE_TOPC:12;
A13: [#](Closed-Interval-TSpace(-1,1)) c= rng g3
proof let x be set;assume x in
[#](Closed-Interval-TSpace(-1,1));
then x in the carrier of (Closed-Interval-TSpace(-1,1));
then A14: x in [.-1,1.] by TOPMETR:25;
then reconsider r=x as Real;
set q=|[r,sqrt(1-r^2)]|;
A15: |.q.|=sqrt((q`1)^2+(q`2)^2) by JGRAPH_3:10
.=sqrt(r^2+(q`2)^2) by EUCLID:56
.=sqrt(r^2+(sqrt(1-r^2))^2) by EUCLID:56;
-1<=r & r<=1 by A14,TOPREAL5:1;
then 1^2>=r^2 by JGRAPH_2:7;
then A16: 1-r^2>=0 by SQUARE_1:12,59;
then A17: 0<=sqrt(1-r^2) by SQUARE_1:def 4;
|.q.|=sqrt(r^2+(1-r^2)) by A15,A16,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then A18: q in P by A1;
q`2=sqrt(1-r^2) by EUCLID:56;
then q in {p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A17,A18
;
then A19: q in dom g3 by A1,A11,Th37;
then g3.q=proj1.q by A2,A4 .=q`1 by PSCOMP_1:def 28.=r by EUCLID:56;
hence x in rng g3 by A19,FUNCT_1:def 5;
end;
reconsider B=[.-1,1.] as non empty Subset of R^1
by TOPMETR:24,TOPREAL5:1;
A20: Closed-Interval-TSpace(-1,1)=R^1|B by TOPMETR:26;
for x,y being set st x in dom g3 & y in dom g3 & g3.x=g3.y holds x=y
proof let x,y be set;assume
A21: x in dom g3 & y in dom g3 & g3.x=g3.y;
then reconsider p1=x as Point of TOP-REAL 2 by A11;
reconsider p2=y as Point of TOP-REAL 2 by A11,A21;
A22: g3.x=proj1.p1 by A2,A4,A21 .=p1`1 by PSCOMP_1:def 28;
A23: g3.y=proj1.p2 by A2,A4,A21 .=p2`1 by PSCOMP_1:def 28;
A24: p1 in P by A6,A11,A21;
p2 in P by A6,A11,A21;
then consider p22 being Point of TOP-REAL 2 such that
A25: p2=p22 & |.p22.|=1 by A1;
1^2= (p2`1)^2+(p2`2)^2 by A25,JGRAPH_3:10;
then A26: 1^2-(p2`1)^2= (p2`2)^2 by XCMPLX_1:26;
consider p11 being Point of TOP-REAL 2 such that
A27: p1=p11 & |.p11.|=1 by A1,A24;
1^2= (p1`1)^2+(p1`2)^2 by A27,JGRAPH_3:10;
then A28: (p1`2)^2 =(p2`2)^2 by A21,A22,A23,A26,XCMPLX_1:26;
p1 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2>=0}
by A1,A11,A21,Th37;
then consider p33 being Point of TOP-REAL 2 such that
A29: p1=p33 & p33 in P & p33`2>=0;
A30: p1`2=sqrt(((p2`2))^2) by A28,A29,SQUARE_1:89;
p2 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2>=0}
by A1,A11,A21,Th37;
then consider p44 being Point of TOP-REAL 2 such that
A31: p2=p44 & p44 in P & p44`2>=0;
(p1`2)=(p2`2) by A30,A31,SQUARE_1:89;
then p1=|[p2`1,p2`2]| by A21,A22,A23,EUCLID:57
.=p2 by EUCLID:57;
hence x=y;
end;
hence proj1|K0 is continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) &
proj1|K0 is one-to-one &
rng (proj1|K0)=[#](Closed-Interval-TSpace(-1,1)) by A3,A12,A13,A20,FUNCT_1:
def 8,JGRAPH_1:63,XBOOLE_0:def 10;
end;
theorem Th43: for P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
ex f being map of
Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Lower_Arc(P)
st f is_homeomorphism & (for q being Point of TOP-REAL 2 st
q in Lower_Arc(P) holds f.(q`1)=q) & f.(-1)=W-min(P) & f.1=E-max(P)
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
set P4=Lower_Arc(P);
A4: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
reconsider g=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Lower_Arc(P);
reconsider g2=g|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
A5: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by JORDAN1:1;
hence g2.p=proj1.p by FUNCT_1:72;
end;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm4;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A6: E-max(P) in Lower_Arc(P) by A4,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A7: W-min(P) in Lower_Arc(P) by A4,XBOOLE_0:def 3;
A8: dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A9: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
then A10: dom g3=K0 by PRE_TOPC:def 10;
A11: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm4;
A12: g3 is one-to-one by A1,Lm4;
K0 is non empty compact by A3,JORDAN5A:1;
then A13: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then g3 is_homeomorphism by A9,A11,A12,A13,COMPTS_1:26;
then A14: g3/" is_homeomorphism by TOPS_2:70;
A15: for q be Point of TOP-REAL 2 st
q in Lower_Arc(P) holds (g3/").(q`1)=q
proof let q be Point of TOP-REAL 2;
assume A16: q in Lower_Arc(P);
reconsider g4=g3 as Function;
A17: q in dom g4 implies q = (g4").(g4.q) & q = (g4"*g4).q
by A12,FUNCT_1:56;
g3.q=proj1.q by A5,A8,A10,A16 .=q`1 by PSCOMP_1:def 28;
hence (g3/").(q`1)=q by A9,A11,A12,A16,A17,PRE_TOPC:def 10,TOPS_2:def 4;
end;
A18: W-min(P)=|[-1,0]| by A1,Th32;
A19: g3/".(-1)=g3/".((|[-1,0]|)`1) by EUCLID:56 .=W-min(P)
by A7,A15,A18;
A20: E-max(P)=|[1,0]| by A1,Th33;
g3/".1=g3/".((|[1,0]|)`1) by EUCLID:56 .=E-max(P) by A6,A15,A20;
hence thesis by A14,A15,A19;
end;
theorem Th44: for P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
ex f being map of
Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Upper_Arc(P)
st f is_homeomorphism & (for q being Point of TOP-REAL 2 st
q in Upper_Arc(P) holds f.(q`1)=q) & f.(-1)=W-min(P) & f.1=E-max(P)
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
set P4=Lower_Arc(P);
A4: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
reconsider g=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Upper_Arc(P);
reconsider g2=g|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
A5: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by JORDAN1:1;
hence g2.p=proj1.p by FUNCT_1:72;
end;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm5;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A6: E-max(P) in Upper_Arc(P) by A4,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A7: W-min(P) in Upper_Arc(P) by A4,XBOOLE_0:def 3;
A8: dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A9: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
then A10: dom g3=K0 by PRE_TOPC:def 10;
A11: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm5;
A12: g3 is one-to-one by A1,Lm5;
K0 is non empty compact by A3,JORDAN5A:1;
then A13: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then g3 is_homeomorphism by A9,A11,A12,A13,COMPTS_1:26;
then A14: g3/" is_homeomorphism by TOPS_2:70;
A15: for q be Point of TOP-REAL 2 st
q in Upper_Arc(P) holds (g3/").(q`1)=q
proof let q be Point of TOP-REAL 2;
assume A16: q in Upper_Arc(P);
reconsider g4=g3 as Function;
A17: q in dom g4 implies q = (g4").(g4.q) & q = (g4"*g4).q
by A12,FUNCT_1:56;
g3.q=proj1.q by A5,A8,A10,A16 .=q`1 by PSCOMP_1:def 28;
hence (g3/").(q`1)=q by A9,A11,A12,A16,A17,PRE_TOPC:def 10,TOPS_2:def 4;
end;
A18: W-min(P)=|[-1,0]| by A1,Th32;
A19: g3/".(-1)=g3/".((|[-1,0]|)`1) by EUCLID:56 .=W-min(P)
by A7,A15,A18;
A20: E-max(P)=|[1,0]| by A1,Th33;
g3/".(1)=g3/".((|[1,0]|)`1) by EUCLID:56 .=E-max(P) by A6,A15,A20;
hence thesis by A14,A15,A19;
end;
theorem Th45: for P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
ex f being map of I[01],(TOP-REAL 2)|Lower_Arc(P)
st f is_homeomorphism & (for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st f.r1=q1 & f.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1>q2`1)&
f.0 = E-max(P) & f.1 = W-min(P)
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then consider f1 being map of
Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Lower_Arc(P)
such that
A2: f1 is_homeomorphism & (for q being Point of TOP-REAL 2 st
q in Lower_Arc(P) holds f1.(q`1)=q)& f1.(-1)=W-min(P) & f1.1=E-max(P)
by Th43;
consider g being map of
I[01],Closed-Interval-TSpace(-1,1) such that
A3: g is_homeomorphism & (for r being Real st r in [.0,1.] holds
g.r=(-2)*r+1)& g.0=1 & g.1=-1 by Th41;
reconsider T= (TOP-REAL 2)|Lower_Arc(P) as non empty TopSpace;
reconsider f2=f1 as map of Closed-Interval-TSpace(-1,1),T;
A4: f2*g is_homeomorphism by A2,A3,TOPS_2:71;
reconsider h=f1*g as map of I[01],(TOP-REAL 2)|Lower_Arc(P);
A5: for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st h.r1=q1 & h.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1>q2`1
proof let q1,q2 be Point of TOP-REAL 2,r1,r2 be Real;
assume A6: h.r1=q1 & h.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.];
A7: now assume A8: r1<r2;
set s1=(-2)*r1+1,s2=(-2)*r2+1;
set p1=|[s1,-sqrt(1-s1^2)]|,p2=|[s2,-sqrt(1-s2^2)]|;
A9: g.r1=(-2)*r1+1 by A3,A6;
A10: g.r2=(-2)*r2+1 by A3,A6;
(-2)*r1 > (-2)*r2 by A8,REAL_1:71;
then A11: (-2)*r1 +1 > (-2)*r2 +1 by REAL_1:67;
r1<=1 by A6,TOPREAL5:1;
then (-2)*r1>=(-2)*1 by REAL_1:52;
then (-2)*r1+1>=(-2)*1+1 by REAL_1:55;
then A12: -1<=s1;
r1>=0 by A6,TOPREAL5:1;
then (-2)*r1<=(-2)*0 by REAL_1:52;
then (-2)*r1+1<=(-2)*0+1 by REAL_1:55;
then s1^2<=1^2 by A12,JGRAPH_2:7;
then A13: 1-s1^2>=0 by SQUARE_1:12,59;
A14: (|[s1,-sqrt(1-s1^2)]|)`1=s1 by EUCLID:56;
A15: (|[s1,-sqrt(1-s1^2)]|)`2=-sqrt(1-s1^2) by EUCLID:56;
sqrt(1-s1^2)>=0 by A13,SQUARE_1:def 4;
then A16: -sqrt(1-s1^2)<=0 by REAL_1:66;
|.p1.|=sqrt((p1`1)^2+(p1`2)^2) by JGRAPH_3:10
.=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A14,A15,SQUARE_1:61
.=sqrt((s1)^2+(1-s1^2)) by A13,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p1 in P by A1;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0}
by A15,A16;
then A17: |[s1,-sqrt(1-s1^2)]| in Lower_Arc(P) by A1,Th38;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r1=f1.s1 by A6,A9,FUNCT_1:22 .=p1 by A2,A14,A17;
then A18: q1`1=s1 by A6,EUCLID:56;
r2<=1 by A6,TOPREAL5:1;
then (-2)*r2>=(-2)*1 by REAL_1:52;
then (-2)*r2+1>=(-2)*1+1 by REAL_1:55;
then A19: -1<=s2;
r2>=0 by A6,TOPREAL5:1;
then (-2)*r2<=(-2)*0 by REAL_1:52;
then (-2)*r2+1<=(-2)*0+1 by REAL_1:55;
then s2^2<=1^2 by A19,JGRAPH_2:7;
then A20: 1-s2^2>=0 by SQUARE_1:12,59;
A21: (|[s2,-sqrt(1-s2^2)]|)`1=s2 by EUCLID:56;
A22: (|[s2,-sqrt(1-s2^2)]|)`2=-sqrt(1-s2^2) by EUCLID:56;
sqrt(1-s2^2)>=0 by A20,SQUARE_1:def 4;
then A23: -sqrt(1-s2^2)<=0 by REAL_1:66;
|.p2.|=sqrt((p2`1)^2+(p2`2)^2) by JGRAPH_3:10
.=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A21,A22,SQUARE_1:61
.=sqrt((s2)^2+(1-s2^2)) by A20,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p2 in P by A1;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0}
by A22,A23;
then A24: |[s2,-sqrt(1-s2^2)]| in Lower_Arc(P) by A1,Th38;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r2=f1.s2 by A6,A10,FUNCT_1:22 .=p2 by A2,A21,A24;
hence q1`1>q2`1 by A6,A11,A18,EUCLID:56;
end;
A25: now assume A26: r2<r1;
set s1=(-2)*r2+1,s2=(-2)*r1+1;
set p1=|[s1,-sqrt(1-s1^2)]|,p2=|[s2,-sqrt(1-s2^2)]|;
A27: g.r2=(-2)*r2+1 by A3,A6;
A28: g.r1=(-2)*r1+1 by A3,A6;
A29: (-2)*r2 > (-2)*r1 by A26,REAL_1:71;
r2<=1 by A6,TOPREAL5:1;
then (-2)*r2>=(-2)*1 by REAL_1:52;
then (-2)*r2+1>=(-2)*1+1 by REAL_1:55;
then A30: -1<=s1;
r2>=0 by A6,TOPREAL5:1;
then (-2)*r2<=(-2)*0 by REAL_1:52;
then (-2)*r2+1<=(-2)*0+1 by REAL_1:55;
then s1^2<=1^2 by A30,JGRAPH_2:7;
then A31: 1-s1^2>=0 by SQUARE_1:12,59;
A32: (|[s1,-sqrt(1-s1^2)]|)`1=s1 by EUCLID:56;
A33: (|[s1,-sqrt(1-s1^2)]|)`2=-sqrt(1-s1^2) by EUCLID:56;
sqrt(1-s1^2)>=0 by A31,SQUARE_1:def 4;
then A34: -sqrt(1-s1^2)<=0 by REAL_1:66;
|.p1.|=sqrt((p1`1)^2+(p1`2)^2) by JGRAPH_3:10
.=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A32,A33,SQUARE_1:61
.=sqrt((s1)^2+(1-s1^2)) by A31,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p1 in P by A1;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0}
by A33,A34;
then A35: |[s1,-sqrt(1-s1^2)]| in Lower_Arc(P) by A1,Th38;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r2=f1.s1 by A6,A27,FUNCT_1:22 .=p1 by A2,A32,A35;
then A36: q2`1=s1 by A6,EUCLID:56;
r1<=1 by A6,TOPREAL5:1;
then (-2)*r1>=(-2)*1 by REAL_1:52;
then (-2)*r1+1>=(-2)*1+1 by REAL_1:55;
then A37: -1<=s2;
r1>=0 by A6,TOPREAL5:1;
then (-2)*r1<=(-2)*0 by REAL_1:52;
then (-2)*r1+1<=(-2)*0+1 by REAL_1:55;
then s2^2<=1^2 by A37,JGRAPH_2:7;
then A38: 1-s2^2>=0 by SQUARE_1:12,59;
A39: (|[s2,-sqrt(1-s2^2)]|)`1=s2 by EUCLID:56;
A40: (|[s2,-sqrt(1-s2^2)]|)`2=-sqrt(1-s2^2) by EUCLID:56;
sqrt(1-s2^2)>=0 by A38,SQUARE_1:def 4;
then A41: -sqrt(1-s2^2)<=0 by REAL_1:66;
|.p2.|=sqrt((p2`1)^2+(p2`2)^2) by JGRAPH_3:10
.=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A39,A40,SQUARE_1:61
.=sqrt((s2)^2+(1-s2^2)) by A38,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p2 in P by A1;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0}
by A40,A41;
then A42: |[s2,-sqrt(1-s2^2)]| in Lower_Arc(P) by A1,Th38;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r1=f1.s2 by A6,A28,FUNCT_1:22 .=p2 by A2,A39,A42;
hence q2`1>q1`1 by A6,A29,A36,A39,REAL_1:67;
end;
now assume A43: q1`1>q2`1;
now assume A44: r1>=r2;
now per cases by A44,REAL_1:def 5;
case r1>r2;
hence contradiction by A25,A43;
case r1=r2;
hence contradiction by A6,A43;
end;
hence contradiction;
end;
hence r1<r2;
end;
hence r1<r2 iff q1`1>q2`1 by A7;
end;
A45: dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then 0 in dom h by TOPREAL5:1;
then A46: h.0=E-max(P) by A2,A3,FUNCT_1:22;
1 in dom h by A45,TOPREAL5:1;
then h.1=W-min(P) by A2,A3,FUNCT_1:22;
hence thesis by A4,A5,A46;
end;
theorem Th46: for P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
ex f being map of I[01],(TOP-REAL 2)|Upper_Arc(P)
st f is_homeomorphism & (for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st f.r1=q1 & f.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1<q2`1)&
f.0 = W-min(P) & f.1 = E-max(P)
proof let P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then consider f1 being map of
Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Upper_Arc(P) such that
A2: f1 is_homeomorphism & (for q being Point of TOP-REAL 2 st
q in Upper_Arc(P) holds f1.(q`1)=q)& f1.(-1)=W-min(P) & f1.1=E-max(P)
by Th44;
consider g being map of
I[01],Closed-Interval-TSpace(-1,1) such that
A3: g is_homeomorphism & (for r being Real st r in [.0,1.] holds
g.r=2*r-1)& g.0=-1 & g.1=1 by Th42;
reconsider T= (TOP-REAL 2)|Upper_Arc(P) as non empty TopSpace;
reconsider f2=f1 as map of Closed-Interval-TSpace(-1,1),T;
A4: f2*g is_homeomorphism by A2,A3,TOPS_2:71;
reconsider h=f1*g as map of I[01],(TOP-REAL 2)|Upper_Arc(P);
A5: for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st h.r1=q1 & h.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1<q2`1
proof let q1,q2 be Point of TOP-REAL 2,r1,r2 be Real;
assume A6: h.r1=q1 & h.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.];
A7: now assume A8: r1>r2;
set s1=2*r1-1,s2=2*r2-1;
set p1=|[s1,sqrt(1-s1^2)]|,p2=|[s2,sqrt(1-s2^2)]|;
A9: g.r1=2*r1-1 by A3,A6;
A10: g.r2=2*r2-1 by A3,A6;
A11: 2*r1 > 2*r2 by A8,REAL_1:70;
r1<=1 by A6,TOPREAL5:1;
then 2*r1<=2*1 by AXIOMS:25;
then A12: 2*r1-1<=2*1-1 by REAL_1:49;
r1>=0 by A6,TOPREAL5:1;
then 2*r1>=2*0 by AXIOMS:25;
then A13: 2*r1-1>=2*0-1 by REAL_1:49;
2*0-1=-1;
then s1^2<=1^2 by A12,A13,JGRAPH_2:7;
then A14: 1-s1^2>=0 by SQUARE_1:12,59;
A15: (|[s1,sqrt(1-s1^2)]|)`1=s1 by EUCLID:56;
A16: (|[s1,sqrt(1-s1^2)]|)`2=sqrt(1-s1^2) by EUCLID:56;
A17: sqrt(1-s1^2)>=0 by A14,SQUARE_1:def 4;
|.p1.|=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A15,A16,JGRAPH_3:10
.=sqrt((s1)^2+(1-s1^2)) by A14,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p1 in P by A1;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0}
by A16,A17;
then A18: |[s1,sqrt(1-s1^2)]| in Upper_Arc(P) by A1,Th37;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r1=f1.s1 by A6,A9,FUNCT_1:22 .=p1 by A2,A15,A18;
then A19: q1`1=s1 by A6,EUCLID:56;
r2<=1 by A6,TOPREAL5:1;
then 2*r2<=2*1 by AXIOMS:25;
then A20: 2*r2-1<=2*1-1 by REAL_1:49;
r2>=0 by A6,TOPREAL5:1;
then 2*r2>=2*0 by AXIOMS:25;
then A21: 2*r2-1>=2*0-1 by REAL_1:49;
2*0-1=-1;
then s2^2<=1^2 by A20,A21,JGRAPH_2:7;
then A22: 1-s2^2>=0 by SQUARE_1:12,59;
A23: (|[s2,sqrt(1-s2^2)]|)`1=s2 by EUCLID:56;
A24: (|[s2,sqrt(1-s2^2)]|)`2=sqrt(1-s2^2) by EUCLID:56;
A25: sqrt(1-s2^2)>=0 by A22,SQUARE_1:def 4;
|.p2.|=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A23,A24,JGRAPH_3:10
.=sqrt((s2)^2+(1-s2^2)) by A22,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p2 in P by A1;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0}
by A24,A25;
then A26: |[s2,sqrt(1-s2^2)]| in Upper_Arc(P) by A1,Th37;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r2=f1.s2 by A6,A10,FUNCT_1:22 .=p2 by A2,A23,A26;
hence q1`1>q2`1 by A6,A11,A19,A23,REAL_1:92;
end;
A27: now assume A28: r2>r1;
set s1=2*r2-1,s2=2*r1-1;
set p1=|[s1,sqrt(1-s1^2)]|,p2=|[s2,sqrt(1-s2^2)]|;
A29: g.r2=2*r2-1 by A3,A6;
A30: g.r1=2*r1-1 by A3,A6;
A31: 2*r2 > 2*r1 by A28,REAL_1:70;
r2<=1 by A6,TOPREAL5:1;
then 2*r2<=2*1 by AXIOMS:25;
then A32: 2*r2-1<=2*1-1 by REAL_1:49;
r2>=0 by A6,TOPREAL5:1;
then 2*r2>=2*0 by AXIOMS:25;
then 2*r2-1>=2*0-1 by REAL_1:49;
then -1<=s1;
then s1^2<=1^2 by A32,JGRAPH_2:7;
then A33: 1-s1^2>=0 by SQUARE_1:12,59;
A34: (|[s1,sqrt(1-s1^2)]|)`1=s1 by EUCLID:56;
A35: (|[s1,sqrt(1-s1^2)]|)`2=sqrt(1-s1^2) by EUCLID:56;
A36: sqrt(1-s1^2)>=0 by A33,SQUARE_1:def 4;
|.p1.|=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A34,A35,JGRAPH_3:10
.=sqrt((s1)^2+(1-s1^2)) by A33,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p1 in P by A1;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0}
by A35,A36;
then A37: |[s1,sqrt(1-s1^2)]| in Upper_Arc(P) by A1,Th37;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r2=f1.s1 by A6,A29,FUNCT_1:22 .=p1 by A2,A34,A37;
then A38: q2`1=s1 by A6,EUCLID:56;
r1<=1 by A6,TOPREAL5:1;
then 2*r1<=2*1 by AXIOMS:25;
then A39: 2*r1-1<=2*1-1 by REAL_1:49;
r1>=0 by A6,TOPREAL5:1;
then 2*r1>=2*0 by AXIOMS:25;
then 2*r1-1>=2*0-1 by REAL_1:49;
then -1<=s2;
then s2^2<=1^2 by A39,JGRAPH_2:7;
then A40: 1-s2^2>=0 by SQUARE_1:12,59;
A41: (|[s2,sqrt(1-s2^2)]|)`1=s2 by EUCLID:56;
A42: (|[s2,sqrt(1-s2^2)]|)`2=sqrt(1-s2^2) by EUCLID:56;
A43: sqrt(1-s2^2)>=0 by A40,SQUARE_1:def 4;
|.p2.|=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A41,A42,JGRAPH_3:10
.=sqrt((s2)^2+(1-s2^2)) by A40,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then p2 in P by A1;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0}
by A42,A43;
then A44: |[s2,sqrt(1-s2^2)]| in Upper_Arc(P) by A1,Th37;
dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then h.r1=f1.s2 by A6,A30,FUNCT_1:22 .=p2 by A2,A41,A44;
hence q2`1>q1`1 by A6,A31,A38,A41,REAL_1:92;
end;
now assume A45: q1`1<q2`1;
now assume A46: r1>=r2;
now per cases by A46,REAL_1:def 5;
case r1>r2;
hence contradiction by A7,A45;
case r1=r2;
hence contradiction by A6,A45;
end;
hence contradiction;
end;
hence r1<r2;
end;
hence r1<r2 iff q1`1<q2`1 by A27;
end;
A47: dom h=[.0,1.] by BORSUK_1:83,FUNCT_2:def 1;
then 0 in dom h by TOPREAL5:1;
then A48: h.0=W-min(P) by A2,A3,FUNCT_1:22;
1 in dom h by A47,TOPREAL5:1;
then h.1=E-max(P) by A2,A3,FUNCT_1:22;
hence thesis by A4,A5,A48;
end;
theorem Th47: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p2 in Upper_Arc(P) & LE p1,p2,P
holds p1 in Upper_Arc(P)
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p2 in Upper_Arc(P) & LE p1,p2,P;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
set P4b=Lower_Arc(P);
A4: P4b is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4b={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4b=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4b,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2
by A2,JORDAN6:def 9;
then E-max(P) in Upper_Arc(P) /\ P4b by TARSKI:def 2;
then A5: E-max(P) in Upper_Arc(P) by XBOOLE_0:def 3;
A6: p1 in Upper_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) or
p1 in Upper_Arc(P) & p2 in Upper_Arc(P) &
LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) or
p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A1,JORDAN6:def 10;
now assume A7: not p1 in Upper_Arc(P);
then p2 in Upper_Arc(P) /\ P4b by A1,A6,XBOOLE_0:def 3;
then A8: p2=E-max(P) by A4,A6,A7,TARSKI:def 2;
then LE p2,p1,Lower_Arc(P),E-max(P),W-min(P) by A3,A6,A7,JORDAN5C:10;
hence contradiction by A3,A5,A6,A7,A8,JORDAN5C:12;
end;
hence p1 in Upper_Arc(P);
end;
theorem Th48: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`1<0 & p2`1<0 & p1`2<0 & p2`2<0
holds p1`1>p2`1 & p1`2<p2`2
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`1<0 & p2`1<0 & p1`2<0 & p2`2<0;
then P is_simple_closed_curve by JGRAPH_3:36;
then A2: p1 in P & p2 in P by A1,JORDAN7:5;
A3: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
now assume p1 in Upper_Arc(P);
then consider p being Point of TOP-REAL 2 such that
A4: p1=p & p in P & p`2>=0 by A3;
thus contradiction by A1,A4;
end;
then A5: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P)
by A1,JORDAN6:def 10;
consider f being map of I[01],(TOP-REAL 2)|Lower_Arc(P)
such that
A6: f is_homeomorphism & (for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st f.r1=q1 & f.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1>q2`1)
& f.0=E-max(P) & f.1=W-min(P) by A1,Th45;
A7: rng f=[#]((TOP-REAL 2)|Lower_Arc(P)) by A6,TOPS_2:def 5
.=Lower_Arc(P) by PRE_TOPC:def 10;
then consider x1 being set such that
A8: x1 in dom f & p1=f.x1 by A5,FUNCT_1:def 5;
consider x2 being set such that
A9: x2 in dom f & p2=f.x2 by A5,A7,FUNCT_1:def 5;
A10: dom f=[#](I[01]) by A6,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
then reconsider r11=x1 as Real by A8;
reconsider r22=x2 as Real by A9,A10;
A11: 0<=r11 & r11<=1 by A8,A10,TOPREAL5:1;
A12: 0<=r22 & r22<=1 by A9,A10,TOPREAL5:1;
A13: r11<r22 iff p1`1>p2`1 by A6,A8,A9,A10;
A14: r11<=r22 by A5,A6,A8,A9,A11,A12,JORDAN5C:def 3;
consider p3 being Point of TOP-REAL 2 such that
A15: p3=p1 & |.p3.|=1 by A1,A2;
1^2=(p1`1)^2+(p1`2)^2 by A15,JGRAPH_3:10;
then 1^2-(p1`1)^2=(p1`2)^2 by XCMPLX_1:26.=(-(p1`2))^2
by SQUARE_1:61;
then A16: 1^2-(-(p1`1))^2=(-(p1`2))^2 by SQUARE_1:61;
A17: -(p1`1)>0 by A1,REAL_1:66;
-(p1`2)>0 by A1,REAL_1:66;
then -(p1`2)=sqrt(1^2-(-(p1`1))^2) by A16,SQUARE_1:89;
then A18: (p1`2)=-sqrt(1^2-(-(p1`1))^2);
consider p4 being Point of TOP-REAL 2 such that
A19: p4=p2 & |.p4.|=1 by A1,A2;
1^2=(p2`1)^2+(p2`2)^2 by A19,JGRAPH_3:10;
then 1^2-(p2`1)^2=(p2`2)^2 by XCMPLX_1:26.=(-(p2`2))^2
by SQUARE_1:61;
then A20: 1^2-(-(p2`1))^2=(-(p2`2))^2 by SQUARE_1:61;
-(p2`2)>0 by A1,REAL_1:66;
then -(p2`2)=sqrt(1^2-(-(p2`1))^2) by A20,SQUARE_1:89;
then A21: (p2`2)=-sqrt(1^2-(-(p2`1))^2);
-(p1`1)< -(p2`1) by A1,A8,A9,A13,A14,REAL_1:50,def 5;
then (-(p1`1))^2 < (-(p2`1))^2 by A17,SQUARE_1:78;
then A22: 1^2- (-(p1`1))^2 > 1^2-(-(p2`1))^2 by REAL_1:92;
1^2-(-(p2`1))^2>=0 by A20,SQUARE_1:72;
then sqrt(1^2- (-(p1`1))^2) > sqrt(1^2-(-(p2`1))^2) by A22,SQUARE_1:95;
hence p1`1>p2`1 & p1`2<p2`2 by A8,A9,A13,A14,A18,A21,REAL_1:50,def 5;
end;
theorem Th49: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`1<0 & p2`1<0 & p1`2>=0 & p2`2>=0
holds p1`1<p2`1 & p1`2<p2`2
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`1<0 & p2`1<0 & p1`2>=0 & p2`2>=0;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P &
LE p1,p2,P & p1<>p2 & p1`1<0 & p2`1<0 & p1`2>=0 & p2`2>=0 by A1,JORDAN7:5;
set P4=Lower_Arc(P);
A4: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
A5: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
A6: now assume p2=W-min(P);
then LE p2,p1,P by A2,A3,JORDAN7:3;
hence contradiction by A1,A2,JORDAN6:72;
end;
now assume A7: p2 in Lower_Arc(P);
p2 in Upper_Arc(P) by A3,A5;
then p2 in {W-min(P),E-max(P)} by A4,A7,XBOOLE_0:def 3;
then A8: p2=W-min(P) or p2=E-max(P) by TARSKI:def 2;
E-max(P)=|[1,0]| by A1,Th33;
then (E-max(P))`1=1 by EUCLID:56;
hence contradiction by A1,A6,A8;
end;
then A9: p1 in Upper_Arc(P) & p2 in Upper_Arc(P)& not p2=W-min(P) &
LE p1,p2,Upper_Arc(P),W-min(P),E-max(P)
by A1,A6,JORDAN6:def 10;
consider f being map of
I[01],(TOP-REAL 2)|Upper_Arc(P)
such that
A10: f is_homeomorphism & (for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st f.r1=q1 & f.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1<q2`1)
& f.0=W-min(P) & f.1=E-max(P) by A1,Th46;
A11: rng f=[#]((TOP-REAL 2)|Upper_Arc(P)) by A10,TOPS_2:def 5
.=Upper_Arc(P) by PRE_TOPC:def 10;
then consider x1 being set such that
A12: x1 in dom f & p1=f.x1 by A9,FUNCT_1:def 5;
consider x2 being set such that
A13: x2 in dom f & p2=f.x2 by A9,A11,FUNCT_1:def 5;
A14: dom f=[#](I[01]) by A10,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
then reconsider r11=x1 as Real by A12;
reconsider r22=x2 as Real by A13,A14;
A15: 0<=r11 & r11<=1 by A12,A14,TOPREAL5:1;
A16: 0<=r22 & r22<=1 by A13,A14,TOPREAL5:1;
A17: r11<r22 iff p1`1<p2`1 by A10,A12,A13,A14;
A18: r11<=r22 by A9,A10,A12,A13,A15,A16,JORDAN5C:def 3;
consider p3 being Point of TOP-REAL 2 such that
A19: p3=p1 & |.p3.|=1 by A3;
1^2=(p1`1)^2+(p1`2)^2 by A19,JGRAPH_3:10;
then 1^2-(p1`1)^2=(p1`2)^2 by XCMPLX_1:26;
then A20: 1^2-(-(p1`1))^2=((p1`2))^2 by SQUARE_1:61;
A21: -(p2`1)>0 by A1,REAL_1:66;
A22: (p1`2)=sqrt(1^2-(-(p1`1))^2) by A1,A20,SQUARE_1:89;
consider p4 being Point of TOP-REAL 2 such that
A23: p4=p2 & |.p4.|=1 by A3;
1^2=(p2`1)^2+(p2`2)^2 by A23,JGRAPH_3:10;
then 1^2-(p2`1)^2=(p2`2)^2 by XCMPLX_1:26;
then 1^2-(-(p2`1))^2=((p2`2))^2 by SQUARE_1:61;
then A24: (p2`2)=sqrt(1^2-(-(p2`1))^2) by A1,SQUARE_1:89;
-(p1`1)> -(p2`1) by A1,A12,A13,A17,A18,REAL_1:50,def 5;
then (-(p1`1))^2 > (-(p2`1))^2 by A21,SQUARE_1:78;
then A25: 1^2- (-(p1`1))^2 < 1^2-(-(p2`1))^2 by REAL_1:92;
1^2=(p1`1)^2+(p1`2)^2 by A19,JGRAPH_3:10;
then 1^2-(p1`1)^2=(p1`2)^2 by XCMPLX_1:26;
then 1^2-(-(p1`1))^2=((p1`2))^2 by SQUARE_1:61;
then 1^2-(-(p1`1))^2>=0 by SQUARE_1:72;
hence p1`1<p2`1 & p1`2<p2`2 by A12,A13,A17,A18,A22,A24,A25,REAL_1:def 5,
SQUARE_1:95;
end;
theorem Th50: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`2>=0 & p2`2>=0
holds p1`1<p2`1
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`2>=0 & p2`2>=0;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P &
LE p1,p2,P & p1<>p2 & p1`2>=0 & p2`2>=0 by A1,JORDAN7:5;
set P4=Lower_Arc(P);
A4: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
A5: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
A6: now assume p2=W-min(P);
then LE p2,p1,P by A2,A3,JORDAN7:3;
hence contradiction by A1,A2,JORDAN6:72;
end;
now assume A7: p2 in Lower_Arc(P);
p2 in Upper_Arc(P) by A3,A5;
then p2 in {W-min(P),E-max(P)} by A4,A7,XBOOLE_0:def 3;
then A8: p2=W-min(P) or p2=E-max(P) by TARSKI:def 2;
consider p8 being Point of TOP-REAL 2 such that
A9: p8=p1 & |.p8.|=1 by A3;
now assume p2=W-min(P);
then LE p2,p1,P by A2,A3,JORDAN7:3;
hence contradiction by A1,A2,JORDAN6:72;
end;
then A10: p2= |[1,0]| by A1,A8,Th33;
then A11: p2`1=1 by EUCLID:56;
A12: now assume A13: p1`1=1;
1^2=(p1`1)^2+(p1`2)^2 by A9,JGRAPH_3:10;
then 1^2-(p1`1)^2=(p1`2)^2 by XCMPLX_1:26;
then 0=((p1`2))^2 by A13,XCMPLX_1:14;
then p1`2=0 by SQUARE_1:73;
hence contradiction by A1,A10,A13,EUCLID:57;
end;
p1`1<=1 by A9,Th3;
hence p1`1<p2`1 by A11,A12,REAL_1:def 5;
end;
then A14: p1 in Upper_Arc(P) & p2 in Upper_Arc(P)& not p2=W-min(P) &
LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) or p1`1<p2`1
by A1,A6,JORDAN6:def 10;
consider f being map of I[01],(TOP-REAL 2)|Upper_Arc(P)
such that
A15: f is_homeomorphism & (for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st f.r1=q1 & f.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1<q2`1)
& f.0=W-min(P) & f.1=E-max(P) by A1,Th46;
A16: rng f=[#]((TOP-REAL 2)|Upper_Arc(P)) by A15,TOPS_2:def 5
.=Upper_Arc(P) by PRE_TOPC:def 10;
now per cases;
case A17: not p1`1 < p2`1;
then consider x1 being set such that
A18: x1 in dom f & p1=f.x1 by A14,A16,FUNCT_1:def 5;
consider x2 being set such that
A19: x2 in dom f & p2=f.x2 by A14,A16,A17,FUNCT_1:def 5;
A20: dom f=[#](I[01]) by A15,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
then reconsider r11=x1 as Real by A18;
reconsider r22=x2 as Real by A19,A20;
A21: 0<=r11 & r11<=1 by A18,A20,TOPREAL5:1;
A22: 0<=r22 & r22<=1 by A19,A20,TOPREAL5:1;
A23: r11<r22 iff p1`1<p2`1 by A15,A18,A19,A20;
r11<=r22 or p1`1<p2`1 by A14,A15,A18,A19,A21,A22,JORDAN5C:def 3;
hence p1`1<p2`1 by A1,A18,A19,A23,REAL_1:def 5;
case p1`1<p2`1;
hence p1`1<p2`1;
end;
hence p1`1<p2`1;
end;
theorem Th51: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`2<=0 & p2`2<=0 & p1<>W-min(P)
holds p1`1>p2`1
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`2<=0 & p2`2<=0 & p1<>W-min(P);
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P &
LE p1,p2,P & p1<>p2 & p1`2<=0 & p2`2<=0 & p1<>W-min(P) by A1,JORDAN7:5;
set P4=Lower_Arc(P);
A4: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
A5: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A1,Th38;
now assume A6: p1 in Upper_Arc(P);
p1 in Lower_Arc(P) by A3,A5;
then p1 in {W-min(P),E-max(P)} by A4,A6,XBOOLE_0:def 3;
then A7: p1=W-min(P) or p1=E-max(P) by TARSKI:def 2;
consider p9 being Point of TOP-REAL 2 such that
A8: p9=p2 & |.p9.|=1 by A3;
A9: p1= |[1,0]| by A1,A7,Th33;
then A10: p1`1=1 by EUCLID:56;
A11: now assume A12: p2`1=1;
1^2 =(p2`1)^2+(p2`2)^2 by A8,JGRAPH_3:10;
then (1)^2-(p2`1)^2=(p2`2)^2 by XCMPLX_1:26;
then 0=((p2`2))^2 by A12,XCMPLX_1:14;
then p2`2=0 by SQUARE_1:73;
hence contradiction by A1,A9,A12,EUCLID:57;
end;
p2`1<=1 by A8,Th3;
hence p1`1>p2`1 by A10,A11,REAL_1:def 5;
end;
then A13: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) or p1`1>p2`1
by A1,JORDAN6:def 10;
consider f being map of I[01],(TOP-REAL 2)|Lower_Arc(P) such that
A14: f is_homeomorphism & (for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st f.r1=q1 & f.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1>q2`1)
& f.0=E-max(P) & f.1=W-min(P) by A1,Th45;
A15: rng f=[#]((TOP-REAL 2)|Lower_Arc(P)) by A14,TOPS_2:def 5
.=Lower_Arc(P) by PRE_TOPC:def 10;
now per cases;
case A16: not p1`1 > p2`1;
then consider x1 being set such that
A17: x1 in dom f & p1=f.x1 by A13,A15,FUNCT_1:def 5;
consider x2 being set such that
A18: x2 in dom f & p2=f.x2 by A13,A15,A16,FUNCT_1:def 5;
A19: dom f=[#](I[01]) by A14,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
then reconsider r11=x1 as Real by A17;
reconsider r22=x2 as Real by A18,A19;
A20: 0<=r11 & r11<=1 by A17,A19,TOPREAL5:1;
A21: 0<=r22 & r22<=1 by A18,A19,TOPREAL5:1;
A22: r11<r22 iff p1`1>p2`1 by A14,A17,A18,A19;
r11<=r22 or p1`1>p2`1 by A13,A14,A17,A18,A20,A21,JORDAN5C:def 3;
hence p1`1>p2`1 by A1,A17,A18,A22,REAL_1:def 5;
case p1`1>p2`1;
hence p1`1>p2`1;
end;
hence p1`1>p2`1;
end;
theorem Th52: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& (p2`2>=0 or p2`1>=0) & LE p1,p2,P
holds p1`2>=0 or p1`1>=0
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& (p2`2>=0 or p2`1>=0) & LE p1,p2,P;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
A4: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P
& (p2`2>=0 or p2`1>=0) & LE p1,p2,P by A1,A2,JORDAN7:5;
A5: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
A6: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A1,Th38;
per cases by A1;
suppose p2`2>=0;
then p2 in Upper_Arc(P) by A4,A5;
then p1 in Upper_Arc(P) by A1,Th47;
then consider p8 being Point of TOP-REAL 2 such that
A7: p8=p1 & p8 in P & p8`2>=0 by A5;
thus p1`2>=0 or p1`1>=0 by A7;
suppose A8: p2`2<0 & p2`1>=0;
then not ex p8 being Point of TOP-REAL 2 st p8=p2 & p8 in P & p8`2>=0;
then A9: not p2 in Upper_Arc(P) by A5;
now per cases by A1,A9,JORDAN6:def 10;
case p1 in Upper_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P);
then consider p8 being Point of TOP-REAL 2 such that
A10: p8=p1 & p8 in P & p8`2>=0 by A5;
thus p1`2>=0 or p1`1>=0 by A10;
case A11: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P)&
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P);
then consider p8 being Point of TOP-REAL 2 such that
A12: p8=p1 & p8 in P & p8`2<=0 by A6;
now assume A13: p1=W-min(P);
then LE p2,p1,Lower_Arc(P),E-max(P),W-min(P) by A3,A11,JORDAN5C:10
;
hence contradiction by A3,A11,A13,JORDAN5C:12;
end;
then p1`1>=p2`1 by A1,A8,A12,Th51;
hence p1`2>=0 or p1`1>=0 by A8;
end;
hence p1`2>=0 or p1`1>=0;
end;
theorem Th53: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`1>=0 & p2`1>=0
holds p1`2>p2`2
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & p1<>p2 & p1`1>=0 & p2`1>=0;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: P={p where p is Point of TOP-REAL 2: |.p.|=1} & p1 in P & p2 in P &
LE p1,p2,P & p1<>p2 & p1`1>=0 & p2`1>=0 by A1,JORDAN7:5;
then consider p4 being Point of TOP-REAL 2 such that
A4: p4=p1 & |.p4.|=1;
consider p3 being Point of TOP-REAL 2 such that
A5: p3=p2 & |.p3.|=1 by A3;
A6: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
A7: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A1,Th38;
W-min(P)=|[-1,0]| by A1,Th32;
then A8:(W-min(P))`2=0 by EUCLID:56;
now per cases;
case A9: p1`2>=0 & p2`2>=0;
then p1`1<p2`1 by A1,Th50;
then (p1`1)^2 < ((p2`1))^2 by A1,SQUARE_1:78;
then A10: 1^2- ((p1`1))^2 > 1^2-((p2`1))^2 by REAL_1:92;
1^2=(p1`1)^2+(p1`2)^2 by A4,JGRAPH_3:10;
then 1^2-((p1`1))^2=((p1`2))^2 by XCMPLX_1:26;
then A11: p1`2=sqrt(1^2-((p1`1))^2) by A9,SQUARE_1:89;
1^2=(p2`1)^2+(p2`2)^2 by A5,JGRAPH_3:10;
then A12: 1^2-((p2`1))^2=((p2`2))^2 by XCMPLX_1:26;
then A13: (p2`2)=sqrt(1^2-((p2`1))^2) by A9,SQUARE_1:89;
1^2-((p2`1))^2>=0 by A12,SQUARE_1:72;
hence p1`2>p2`2 by A10,A11,A13,SQUARE_1:95;
case p1`2>=0 & p2`2<0;
hence p1`2>p2`2;
case A14: p1`2<0 & p2`2>=0;
then A15: p1 in Lower_Arc(P) by A3,A7;
p2 in Upper_Arc(P) by A3,A6,A14;
then LE p2,p1,P by A8,A14,A15,JORDAN6:def 10;
hence contradiction by A1,A2,JORDAN6:72;
case A16: p1`2<0 & p2`2<0;
then not ex p being Point of TOP-REAL 2 st p=p1 & p in P & p`2>=0;
then not p1 in Upper_Arc(P) by A6;
then A17: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A1,JORDAN6:def 10;
consider f being map of I[01],(TOP-REAL 2)|Lower_Arc(P) such that
A18: f is_homeomorphism & (for q1,q2 being Point of TOP-REAL 2,
r1,r2 being Real st f.r1=q1 & f.r2=q2 &
r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1>q2`1)
& f.0=E-max(P) & f.1=W-min(P) by A1,Th45;
A19: rng f=[#]((TOP-REAL 2)|Lower_Arc(P)) by A18,TOPS_2:def 5
.=Lower_Arc(P) by PRE_TOPC:def 10;
then consider x1 being set such that
A20: x1 in dom f & p1=f.x1 by A17,FUNCT_1:def 5;
consider x2 being set such that
A21: x2 in dom f & p2=f.x2 by A17,A19,FUNCT_1:def 5;
A22: dom f=[#](I[01]) by A18,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
then reconsider r11=x1 as Real by A20;
reconsider r22=x2 as Real by A21,A22;
A23: 0<=r11 & r11<=1 by A20,A22,TOPREAL5:1;
A24: 0<=r22 & r22<=1 by A21,A22,TOPREAL5:1;
A25: r11<r22 iff p1`1>p2`1 by A18,A20,A21,A22;
A26: r11<=r22 by A17,A18,A20,A21,A23,A24,JORDAN5C:def 3;
consider p3 being Point of TOP-REAL 2 such that
A27: p3=p1 & |.p3.|=1 by A3;
1^2=(p1`1)^2+(p1`2)^2 by A27,JGRAPH_3:10;
then A28: 1^2-(p1`1)^2=(p1`2)^2 by XCMPLX_1:26;
then A29: 1^2-((p1`1))^2=(-(p1`2))^2 by SQUARE_1:61;
-(p1`2)>0 by A16,REAL_1:66;
then A30: -(p1`2)=sqrt(1^2-((p1`1))^2) by A29,SQUARE_1:89;
consider p4 being Point of TOP-REAL 2 such that
A31: p4=p2 & |.p4.|=1 by A3;
1^2=(p2`1)^2+(p2`2)^2 by A31,JGRAPH_3:10;
then 1^2-(p2`1)^2=(p2`2)^2 by XCMPLX_1:26;
then A32: 1^2-((p2`1))^2=(-(p2`2))^2 by SQUARE_1:61;
-(p2`2)>0 by A16,REAL_1:66;
then A33: -(p2`2)=sqrt(1^2-((p2`1))^2) by A32,SQUARE_1:89;
((p1`1))^2 > ((p2`1))^2 by A1,A20,A21,A25,A26,REAL_1:def 5,SQUARE_1:78;
then A34: 1^2- ((p1`1))^2 < 1^2-((p2`1))^2 by REAL_1:92;
1^2-((p1`1))^2>=0 by A28,SQUARE_1:72;
then sqrt(1^2- ((p1`1))^2) < sqrt(1^2-((p2`1))^2) by A34,SQUARE_1:95;
hence p1`2>p2`2 by A30,A33,REAL_1:50;
end;
hence p1`2>p2`2;
end;
theorem Th54: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P
& p1`1<0 & p2`1<0 & p1`2<0 & p2`2<0 &
(p1`1>=p2`1 or p1`2<=p2`2) holds LE p1,p2,P
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p1`1<0 & p2`1<0 & p1`2<0 & p2`2<0 &
(p1`1>=p2`1 or p1`2<=p2`2);
then consider p3 being Point of TOP-REAL 2 such that
A2: p3=p1 & |.p3.|=1;
consider p3 being Point of TOP-REAL 2 such that
A3: p3=p2 & |.p3.|=1 by A1;
A4: -p2`2>0 by A1,REAL_1:66;
A5: now assume p1`2<=p2`2;
then -p1`2>=-p2`2 by REAL_1:50;
then (-(p1`2))^2 >= (-(p2`2))^2 by A4,SQUARE_1:77;
then A6: 1^2- (-(p1`2))^2 <= 1^2-(-(p2`2))^2 by REAL_1:92;
1^2=(p1`1)^2+(p1`2)^2 by A2,JGRAPH_3:10;
then 1^2-(p1`2)^2=(p1`1)^2 by XCMPLX_1:26;
then A7: 1^2-(-(p1`2))^2=((p1`1))^2 by SQUARE_1:61;
then A8: 1^2-(-(p1`2))^2=(-(p1`1))^2 by SQUARE_1:61;
-(p1`1)>=0 by A1,REAL_1:66;
then A9: -(p1`1)=sqrt(1^2-(-(p1`2))^2) by A8,SQUARE_1:89;
1^2=(p2`1)^2+(p2`2)^2 by A3,JGRAPH_3:10;
then 1^2-(p2`2)^2=(p2`1)^2 by XCMPLX_1:26;
then 1^2-(-(p2`2))^2=((p2`1))^2 by SQUARE_1:61;
then A10: 1^2-(-(p2`2))^2=(-(p2`1))^2 by SQUARE_1:61;
-(p2`1)>=0 by A1,REAL_1:66;
then A11: -(p2`1)=sqrt(1^2-(-(p2`2))^2) by A10,SQUARE_1:89;
1^2-(-(p1`2))^2>=0 by A7,SQUARE_1:72;
then sqrt(1^2- (-(p1`2))^2) <= sqrt(1^2-(-(p2`2))^2) by A6,SQUARE_1:94;
hence p1`1>=p2`1 by A9,A11,REAL_1:50;
end;
A12: P is_simple_closed_curve by A1,JGRAPH_3:36;
then A13: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
set P4=Lower_Arc(P);
A14: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A12,JORDAN6:def 9;
A15: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
A16: now assume not p1 in Lower_Arc(P);
then p1 in Upper_Arc(P) by A1,A14,XBOOLE_0:def 2;
then consider p being Point of TOP-REAL 2 such that
A17: p1=p & p in P & p`2>=0 by A15;
thus contradiction by A1,A17;
end;
A18: now assume not p2 in Lower_Arc(P);
then p2 in Upper_Arc(P) by A1,A14,XBOOLE_0:def 2;
then consider p being Point of TOP-REAL 2 such that
A19: p2=p & p in P & p`2>=0 by A15;
thus contradiction by A1,A19;
end;
A20: W-min(P)=|[-1,0]| by A1,Th32;
A21: E-max(P)=|[1,0]| by A1,Th33;
A22: now assume A23: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th32;
hence contradiction by A1,A23,EUCLID:56;
end;
for g being map of I[01], (TOP-REAL 2)|P4,
s1, s2 being Real st g is_homeomorphism
& g.0 = E-max(P) & g.1 = W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof let g be map of I[01], (TOP-REAL 2)|P4,
s1, s2 be Real;
assume A24: g is_homeomorphism
& g.0 = E-max(P) & g.1 =W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1;
then A25: dom g=[#](I[01]) by TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
reconsider g0=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Lower_Arc(P);
reconsider g2=g0|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm4;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A26: E-max(P) in Lower_Arc(P) by A14,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A27: W-min(P) in Lower_Arc(P) by A14,XBOOLE_0:def 3;
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A28: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
A29: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm4;
A30: g3 is one-to-one by A1,Lm4;
K0 is non empty compact by A13,JORDAN5A:1;
then A31: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then A32: g3 is_homeomorphism by A28,A29,A30,A31,COMPTS_1:26;
reconsider h=g3*g as map
of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(-1,1)
by TOPMETR:27;
A33: h is_homeomorphism by A24,A32,TOPMETR:27,TOPS_2:71;
A34: 0 in dom g by A25,TOPREAL5:1;
A35: 1 in dom g by A25,TOPREAL5:1;
A36: s1 in [.0,1.] by A24,TOPREAL5:1;
A37: s2 in [.0,1.] by A24,TOPREAL5:1;
A38: -1=(|[-1,0]|)`1 by EUCLID:56.=proj1.(|[-1,0]|) by PSCOMP_1:def 28
.=g3.(g.1) by A20,A24,A27,FUNCT_1:72
.= h.1 by A35,FUNCT_1:23;
A39: 1=(|[1,0]|)`1 by EUCLID:56.=g0.(|[1,0]|) by PSCOMP_1:def 28
.=g3.(|[1,0]|) by A21,A26,FUNCT_1:72 .= h.0 by A21,A24,A34,FUNCT_1:23;
A40: p1`1=g0.p1 by PSCOMP_1:def 28
.=g3.(g.s1) by A16,A24,FUNCT_1:72
.= h.s1 by A25,A36,FUNCT_1:23;
p2`1=proj1.p2 by PSCOMP_1:def 28
.=g3.(g.s2) by A18,A24,FUNCT_1:72
.= h.s2 by A25,A37,FUNCT_1:23;
hence s1 <= s2 by A1,A5,A33,A36,A37,A38,A39,A40,Th12;
end;
then p1 in Lower_Arc(P) & p2 in Lower_Arc(P) & not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P)
by A16,A18,A22,JORDAN5C:def 3;
hence LE p1,p2,P by JORDAN6:def 10;
end;
theorem for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P
& p1`1>0 & p2`1>0 & p1`2<0 & p2`2<0 &
(p1`1>=p2`1 or p1`2>=p2`2) holds LE p1,p2,P
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P
& p1`1>0 & p2`1>0 & p1`2<0 & p2`2<0 &
(p1`1>=p2`1 or p1`2>=p2`2);
then consider p3 being Point of TOP-REAL 2 such that
A2: p3=p1 & |.p3.|=1;
consider p3 being Point of TOP-REAL 2 such that
A3: p3=p2 & |.p3.|=1 by A1;
A4: -p1`2>0 by A1,REAL_1:66;
A5: now assume p1`2>=p2`2;
then -p1`2<=-p2`2 by REAL_1:50;
then (-(p1`2))^2 <= (-(p2`2))^2 by A4,SQUARE_1:77;
then A6: 1^2- (-(p1`2))^2 >= 1^2-(-(p2`2))^2 by REAL_1:92;
1^2=(p1`1)^2+(p1`2)^2 by A2,JGRAPH_3:10;
then 1^2-(p1`2)^2=(p1`1)^2 by XCMPLX_1:26;
then 1^2-(-(p1`2))^2=((p1`1))^2 by SQUARE_1:61;
then A7: (p1`1)=sqrt(1^2-(-(p1`2))^2) by A1,SQUARE_1:89;
1^2=(p2`1)^2+(p2`2)^2 by A3,JGRAPH_3:10;
then 1^2-(p2`2)^2=(p2`1)^2 by XCMPLX_1:26;
then A8: 1^2-(-(p2`2))^2=((p2`1))^2 by SQUARE_1:61;
then A9: (p2`1)=sqrt(1^2-(-(p2`2))^2) by A1,SQUARE_1:89;
1^2-(-(p2`2))^2>=0 by A8,SQUARE_1:72;
hence p1`1>=p2`1 by A6,A7,A9,SQUARE_1:94;
end;
A10: P is_simple_closed_curve by A1,JGRAPH_3:36;
then A11: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
set P4=Lower_Arc(P);
A12: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A10,JORDAN6:def 9;
A13: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
A14: now assume not p1 in Lower_Arc(P);
then p1 in Upper_Arc(P) by A1,A12,XBOOLE_0:def 2;
then consider p being Point of TOP-REAL 2 such that
A15: p1=p & p in P & p`2>=0 by A13;
thus contradiction by A1,A15;
end;
A16: now assume not p2 in Lower_Arc(P);
then p2 in Upper_Arc(P) by A1,A12,XBOOLE_0:def 2;
then consider p being Point of TOP-REAL 2 such that
A17: p2=p & p in P & p`2>=0 by A13;
thus contradiction by A1,A17;
end;
A18: W-min(P)=|[-1,0]| by A1,Th32;
A19: E-max(P)=|[1,0]| by A1,Th33;
A20: now assume A21: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th32;
hence contradiction by A1,A21,EUCLID:56;
end;
for g being map of I[01], (TOP-REAL 2)|P4,
s1, s2 being Real st
g is_homeomorphism
& g.0 = E-max(P) & g.1 = W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof let g be map of I[01], (TOP-REAL 2)|P4,
s1, s2 be Real;
assume A22: g is_homeomorphism
& g.0 = E-max(P) & g.1 =W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1;
then A23: dom g=[#](I[01]) by TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
reconsider g0=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Lower_Arc(P);
reconsider g2=g0|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm4;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A24: E-max(P) in Lower_Arc(P) by A12,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A25: W-min(P) in Lower_Arc(P) by A12,XBOOLE_0:def 3;
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A26: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
A27: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm4;
A28: g3 is one-to-one by A1,Lm4;
K0 is non empty compact by A11,JORDAN5A:1;
then A29: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then A30: g3 is_homeomorphism by A26,A27,A28,A29,COMPTS_1:26;
reconsider h=g3*g as map
of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(-1,1)
by TOPMETR:27;
A31: h is_homeomorphism by A22,A30,TOPMETR:27,TOPS_2:71;
A32: 0 in dom g by A23,TOPREAL5:1;
A33: 1 in dom g by A23,TOPREAL5:1;
A34: s1 in [.0,1.] by A22,TOPREAL5:1;
A35: s2 in [.0,1.] by A22,TOPREAL5:1;
A36: -1=(|[-1,0]|)`1 by EUCLID:56.=proj1.(|[-1,0]|) by PSCOMP_1:def 28
.=g3.(g.1) by A18,A22,A25,FUNCT_1:72
.= h.1 by A33,FUNCT_1:23;
A37: 1=(|[1,0]|)`1 by EUCLID:56.=proj1.(|[1,0]|) by PSCOMP_1:def 28
.=g3.(g.0) by A19,A22,A24,FUNCT_1:72
.= h.0 by A32,FUNCT_1:23;
A38: p1`1=g0.p1 by PSCOMP_1:def 28
.=g3.(g.s1) by A14,A22,FUNCT_1:72
.= h.s1 by A23,A34,FUNCT_1:23;
p2`1=proj1.p2 by PSCOMP_1:def 28
.=g3.p2 by A16,FUNCT_1:72 .= h.s2 by A22,A23,A35,FUNCT_1:23;
hence s1 <= s2 by A1,A5,A31,A34,A35,A36,A37,A38,Th12;
end;
then p1 in Lower_Arc(P) & p2 in Lower_Arc(P) & not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P)
by A14,A16,A20,JORDAN5C:def 3;
hence LE p1,p2,P by JORDAN6:def 10;
end;
theorem Th56: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P
& p1`1<0 & p2`1<0 & p1`2>=0 & p2`2>=0 &
(p1`1<=p2`1 or p1`2<=p2`2) holds LE p1,p2,P
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P
& p1`1<0 & p2`1<0 & p1`2>=0 & p2`2>=0 &
(p1`1<=p2`1 or p1`2<=p2`2);
then consider p3 being Point of TOP-REAL 2 such that
A2: p3=p1 & |.p3.|=1;
consider p3 being Point of TOP-REAL 2 such that
A3: p3=p2 & |.p3.|=1 by A1;
A4: now assume p1`2<=p2`2;
then ((p1`2))^2 <= ((p2`2))^2 by A1,SQUARE_1:77;
then A5: 1^2- ((p1`2))^2 >= 1^2-((p2`2))^2 by REAL_1:92;
1^2=(p1`1)^2+(p1`2)^2 by A2,JGRAPH_3:10;
then 1^2-(p1`2)^2=(p1`1)^2 by XCMPLX_1:26;
then A6: 1^2-(p1`2)^2=(-(p1`1))^2 by SQUARE_1:61;
-(p1`1)>=0 by A1,REAL_1:66;
then A7: -(p1`1)=sqrt(1^2-((p1`2))^2) by A6,SQUARE_1:89;
1^2=(p2`1)^2+(p2`2)^2 by A3,JGRAPH_3:10;
then A8: 1^2-(p2`2)^2=(p2`1)^2 by XCMPLX_1:26;
then A9: 1^2-(p2`2)^2=(-(p2`1))^2 by SQUARE_1:61;
-(p2`1)>=0 by A1,REAL_1:66;
then A10: -(p2`1)=sqrt(1^2-((p2`2))^2) by A9,SQUARE_1:89;
1^2-((p2`2))^2>=0 by A8,SQUARE_1:72;
then sqrt(1^2- ((p1`2))^2) >= sqrt(1^2-((p2`2))^2) by A5,SQUARE_1:94;
hence p1`1<=p2`1 by A7,A10,REAL_1:50;
end;
A11: P is_simple_closed_curve by A1,JGRAPH_3:36;
then A12: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
set P4=Lower_Arc(P);
set P4b=Upper_Arc(P);
A13: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A11,JORDAN6:def 9;
A14: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
then A15: p1 in Upper_Arc(P) by A1;
A16: p2 in Upper_Arc(P) by A1,A14;
A17: W-min(P)=|[-1,0]| by A1,Th32;
A18: E-max(P)=|[1,0]| by A1,Th33;
for g being map of I[01], (TOP-REAL 2)|P4b, s1, s2 being Real st
g is_homeomorphism
& g.0 = W-min(P) & g.1 = E-max(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof let g be map of I[01], (TOP-REAL 2)|P4b,
s1, s2 be Real;
assume A19: g is_homeomorphism
& g.0 = W-min(P) & g.1 = E-max(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1;
then A20: dom g=[#](I[01]) by TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
reconsider g0=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Upper_Arc(P);
reconsider g2=g0|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm5;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A21: E-max(P) in Upper_Arc(P) by A13,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A22: W-min(P) in Upper_Arc(P) by A13,XBOOLE_0:def 3;
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A23: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
A24: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm5;
A25: g3 is one-to-one by A1,Lm5;
K0 is non empty compact by A12,JORDAN5A:1;
then A26: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then A27: g3 is_homeomorphism by A23,A24,A25,A26,COMPTS_1:26;
reconsider h=g3*g as map
of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(-1,1)
by TOPMETR:27;
A28: h is_homeomorphism by A19,A27,TOPMETR:27,TOPS_2:71;
A29: 0 in dom g by A20,TOPREAL5:1;
A30: 1 in dom g by A20,TOPREAL5:1;
A31: s1 in [.0,1.] by A19,TOPREAL5:1;
A32: s2 in [.0,1.] by A19,TOPREAL5:1;
A33: -1=(|[-1,0]|)`1 by EUCLID:56.=proj1.(|[-1,0]|) by PSCOMP_1:def 28
.=g3.(g.0) by A17,A19,A22,FUNCT_1:72
.= h.0 by A29,FUNCT_1:23;
A34: 1=(|[1,0]|)`1 by EUCLID:56.=g0.(|[1,0]|) by PSCOMP_1:def 28
.=g3.(|[1,0]|) by A18,A21,FUNCT_1:72 .= h.1 by A18,A19,A30,FUNCT_1:23;
A35: p1`1=g0.p1 by PSCOMP_1:def 28
.=g3.(g.s1) by A15,A19,FUNCT_1:72
.= h.s1 by A20,A31,FUNCT_1:23;
p2`1=g0.p2 by PSCOMP_1:def 28
.=g3.p2 by A16,FUNCT_1:72 .= h.s2 by A19,A20,A32,FUNCT_1:23;
hence s1 <= s2 by A1,A4,A28,A31,A32,A33,A34,A35,Th11;
end;
then p1 in Upper_Arc(P) & p2 in Upper_Arc(P) &
LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) by A15,A16,JORDAN5C:def 3;
hence LE p1,p2,P by JORDAN6:def 10;
end;
theorem Th57: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P
& p1`2>=0 & p2`2>=0 &
p1`1<=p2`1 holds LE p1,p2,P
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p1`2>=0 & p2`2>=0 & p1`1<=p2`1;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
set P4=Lower_Arc(P);
set P4b=Upper_Arc(P);
A4: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
A5: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
then A6: p1 in Upper_Arc(P) by A1;
A7: p2 in Upper_Arc(P) by A1,A5;
A8: W-min(P)=|[-1,0]| by A1,Th32;
A9: E-max(P)=|[1,0]| by A1,Th33;
for g being map of I[01], (TOP-REAL 2)|P4b,
s1, s2 being Real st
g is_homeomorphism
& g.0 = W-min(P) & g.1 = E-max(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof let g be map of I[01], (TOP-REAL 2)|P4b,
s1, s2 be Real;
assume A10: g is_homeomorphism
& g.0 = W-min(P) & g.1 = E-max(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1;
then A11: dom g=[#](I[01]) by TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
reconsider g0=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Upper_Arc(P);
reconsider g2=g0|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm5;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A12: E-max(P) in Upper_Arc(P) by A4,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A13: W-min(P) in Upper_Arc(P) by A4,XBOOLE_0:def 3;
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A14: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
A15: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm5;
A16: g3 is one-to-one by A1,Lm5;
K0 is non empty compact by A3,JORDAN5A:1;
then A17: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then A18: g3 is_homeomorphism by A14,A15,A16,A17,COMPTS_1:26;
reconsider h=g3*g as map
of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(-1,1)
by TOPMETR:27;
A19: h is_homeomorphism by A10,A18,TOPMETR:27,TOPS_2:71;
A20: 0 in dom g by A11,TOPREAL5:1;
A21: 1 in dom g by A11,TOPREAL5:1;
A22: s1 in [.0,1.] by A10,TOPREAL5:1;
A23: s2 in [.0,1.] by A10,TOPREAL5:1;
A24: -1=(|[-1,0]|)`1 by EUCLID:56.=proj1.(|[-1,0]|) by PSCOMP_1:def 28
.=g3.(|[-1,0]|) by A8,A13,FUNCT_1:72.= h.0 by A8,A10,A20,FUNCT_1:23;
A25: 1=(|[1,0]|)`1 by EUCLID:56.=g0.(|[1,0]|) by PSCOMP_1:def 28
.=g3.(|[1,0]|) by A9,A12,FUNCT_1:72
.= h.1 by A9,A10,A21,FUNCT_1:23;
A26: p1`1=g0.p1 by PSCOMP_1:def 28
.=g3.p1 by A6,FUNCT_1:72 .= h.s1 by A10,A11,A22,FUNCT_1:23;
p2`1=g0.p2 by PSCOMP_1:def 28
.=g3.p2 by A7,FUNCT_1:72 .= h.s2 by A10,A11,A23,FUNCT_1:23;
hence s1 <= s2 by A1,A19,A22,A23,A24,A25,A26,Th11;
end;
then p1 in Upper_Arc(P) & p2 in Upper_Arc(P) &
LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) by A6,A7,JORDAN5C:def 3;
hence LE p1,p2,P by JORDAN6:def 10;
end;
theorem Th58: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p1`1>=0 & p2`1>=0 &
p1`2>=p2`2 holds LE p1,p2,P
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p1`1>=0 & p2`1>=0 & p1`2>=p2`2;
then consider p3 being Point of TOP-REAL 2 such that
A2: p3=p1 & |.p3.|=1;
consider p3 being Point of TOP-REAL 2 such that
A3: p3=p2 & |.p3.|=1 by A1;
A4: P is_simple_closed_curve by A1,JGRAPH_3:36;
then A5: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
set P4b=Lower_Arc(P);
A6: P4b is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4b={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4b=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4b,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A4,JORDAN6:def 9;
A7: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A1,Th37;
A8: W-min(P)=|[-1,0]| by A1,Th32;
A9: E-max(P)=|[1,0]| by A1,Th33;
now per cases;
case p1 in Upper_Arc(P) & p2 in Upper_Arc(P);
then consider p22 being Point of TOP-REAL 2 such that
A10: p2=p22 & p22 in P & p22`2>=0 by A7;
((p1`2))^2 >= ((p2`2))^2 by A1,A10,SQUARE_1:77;
then A11: 1^2- ((p1`2))^2 <= 1^2-((p2`2))^2 by REAL_1:92;
1^2=(p1`1)^2+(p1`2)^2 by A2,JGRAPH_3:10;
then A12: 1^2-((p1`2))^2=((p1`1))^2 by XCMPLX_1:26;
then A13: (p1`1)=sqrt(1^2-((p1`2))^2) by A1,SQUARE_1:89;
1^2=(p2`1)^2+(p2`2)^2 by A3,JGRAPH_3:10;
then 1^2-((p2`2))^2=((p2`1))^2 by XCMPLX_1:26;
then A14: (p2`1)=sqrt(1^2-((p2`2))^2) by A1,SQUARE_1:89;
1^2-((p1`2))^2>=0 by A12,SQUARE_1:72;
then sqrt(1^2- ((p1`2))^2) <= sqrt(1^2-((p2`2))^2) by A11,SQUARE_1:94;
hence LE p1,p2,P by A1,A10,A13,A14,Th57;
case A15: p1 in Upper_Arc(P) & not p2 in Upper_Arc(P);
then A16: p2 in Lower_Arc(P) by A1,A6,XBOOLE_0:def 2;
now assume A17: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th32;
then p2`2=0 by A17,EUCLID:56;
hence contradiction by A1,A7,A15;
end;
hence LE p1,p2,P by A15,A16,JORDAN6:def 10;
case A18:not p1 in Upper_Arc(P) & p2 in Upper_Arc(P);
then consider p9 being Point of TOP-REAL 2 such that
A19: p2=p9 & p9 in P & p9`2>=0 by A7;
thus contradiction by A1,A7,A18,A19;
case A20: not p1 in Upper_Arc(P) & not p2 in Upper_Arc(P);
then A21: p1 in Lower_Arc(P) by A1,A6,XBOOLE_0:def 2;
A22: p2 in Lower_Arc(P) by A1,A6,A20,XBOOLE_0:def 2;
p1`2<0 by A1,A7,A20;
then A23: -p1`2>0 by REAL_1:66;
-p1`2<=-p2`2 by A1,REAL_1:50;
then (-(p1`2))^2 <= (-(p2`2))^2 by A23,SQUARE_1:77;
then A24: 1^2- (-(p1`2))^2 >= 1^2-(-(p2`2))^2 by REAL_1:92;
1^2=(p1`1)^2+(p1`2)^2 by A2,JGRAPH_3:10;
then 1^2-(p1`2)^2=(p1`1)^2 by XCMPLX_1:26;
then 1^2-(-(p1`2))^2=((p1`1))^2 by SQUARE_1:61;
then A25: p1`1=sqrt(1^2-(-(p1`2))^2) by A1,SQUARE_1:89;
1^2=(p2`1)^2+(p2`2)^2 by A3,JGRAPH_3:10;
then 1^2-(p2`2)^2=(p2`1)^2 by XCMPLX_1:26;
then A26: 1^2-(-(p2`2))^2=((p2`1))^2 by SQUARE_1:61;
then A27: (p2`1)=sqrt(1^2-(-(p2`2))^2) by A1,SQUARE_1:89;
1^2-(-(p2`2))^2>=0 by A26,SQUARE_1:72;
then A28: p1`1>=p2`1 by A24,A25,A27,SQUARE_1:94;
A29: for g being map of I[01], (TOP-REAL 2)|P4b,
s1, s2 being Real st
g is_homeomorphism
& g.0 = E-max(P) & g.1 = W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof let g be map of I[01], (TOP-REAL 2)|P4b, s1, s2 be Real;
assume A30: g is_homeomorphism
& g.0 = E-max(P) & g.1 = W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1;
then A31: dom g=[#](I[01]) by TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
reconsider g0=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Lower_Arc(P);
reconsider g2=g0|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm4;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A32: E-max(P) in Lower_Arc(P) by A6,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A33: W-min(P) in Lower_Arc(P) by A6,XBOOLE_0:def 3;
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A34: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
A35: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm4;
A36: g3 is one-to-one by A1,Lm4;
K0 is non empty compact by A5,JORDAN5A:1;
then A37: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then A38: g3 is_homeomorphism by A34,A35,A36,A37,COMPTS_1:26;
reconsider h=g3*g as map
of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(-1,1)
by TOPMETR:27;
A39: h is_homeomorphism by A30,A38,TOPMETR:27,TOPS_2:71;
A40: 0 in dom g by A31,TOPREAL5:1;
A41: 1 in dom g by A31,TOPREAL5:1;
A42: s1 in [.0,1.] by A30,TOPREAL5:1;
A43: s2 in [.0,1.] by A30,TOPREAL5:1;
A44: -1=(|[-1,0]|)`1 by EUCLID:56.=proj1.(|[-1,0]|) by PSCOMP_1:def 28
.=g3.(|[-1,0]|) by A8,A33,FUNCT_1:72 .= h.1 by A8,A30,A41,FUNCT_1:23
;
A45: 1=(|[1,0]|)`1 by EUCLID:56.=proj1.(|[1,0]|) by PSCOMP_1:def 28
.=g3.(|[1,0]|) by A9,A32,FUNCT_1:72 .= h.0 by A9,A30,A40,FUNCT_1:23;
A46: p1`1=g0.p1 by PSCOMP_1:def 28
.=g3.p1 by A21,FUNCT_1:72 .= h.s1 by A30,A31,A42,FUNCT_1:23;
p2`1=g0.p2 by PSCOMP_1:def 28
.=g3.p2 by A22,FUNCT_1:72 .= h.s2 by A30,A31,A43,FUNCT_1:23;
hence s1 <= s2 by A28,A39,A42,A43,A44,A45,A46,Th12;
end;
A47: now assume A48: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th32;
then p2`2=0 by A48,EUCLID:56;
hence contradiction by A1,A7,A20;
end;
p1 in Lower_Arc(P) & p2 in Lower_Arc(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A21,A22,A29,JORDAN5C:def 3;
hence LE p1,p2,P by A47,JORDAN6:def 10;
end;
hence LE p1,p2,P;
end;
theorem Th59: for p1,p2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p1`2<=0 & p2`2<=0 & p2<>W-min(P) &
p1`1>=p2`1 holds LE p1,p2,P
proof let p1,p2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p1`2<=0 & p2`2<=0 & p2<>W-min(P) &
p1`1>=p2`1;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
set P4=Lower_Arc(P);
A4: P4 is_an_arc_of E-max(P),W-min(P)
& Upper_Arc(P) /\ P4={W-min(P),E-max(P)}
& Upper_Arc(P) \/ P4=P
& First_Point(Upper_Arc(P),W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2>
Last_Point(P4,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 by A2,JORDAN6:def 9;
A5: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A1,Th38;
then A6: p1 in Lower_Arc(P) by A1;
A7: p2 in Lower_Arc(P) by A1,A5;
A8: W-min(P)=|[-1,0]| by A1,Th32;
A9: E-max(P)=|[1,0]| by A1,Th33;
for g being map of I[01], (TOP-REAL 2)|P4, s1, s2 being Real st
g is_homeomorphism
& g.0 = E-max(P) & g.1 = W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof let g be map of I[01], (TOP-REAL 2)|P4, s1, s2 be Real;
assume A10: g is_homeomorphism
& g.0 = E-max(P) & g.1 = W-min(P)
& g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1;
then A11: dom g=[#](I[01]) by TOPS_2:def 5
.=[.0,1.] by BORSUK_1:83,PRE_TOPC:12;
reconsider g0=proj1 as map of TOP-REAL 2,R^1 by JGRAPH_2:16;
set K0=Lower_Arc(P);
reconsider g2=g0|K0 as map of (TOP-REAL 2)|K0,R^1 by JGRAPH_3:12;
reconsider g3=g2 as continuous map of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm4;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A12: E-max(P) in Lower_Arc(P) by A4,XBOOLE_0:def 3;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then A13: W-min(P) in Lower_Arc(P) by A4,XBOOLE_0:def 3;
dom g3=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then A14: dom g3=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12;
A15: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A1,Lm4;
A16: g3 is one-to-one by A1,Lm4;
K0 is non empty compact by A3,JORDAN5A:1;
then A17: (TOP-REAL 2)|K0 is compact by COMPTS_1:12;
Closed-Interval-TSpace(-1,1)
=TopSpaceMetr(Closed-Interval-MSpace(-1,1)) by TOPMETR:def 8;
then Closed-Interval-TSpace(-1,1) is_T2 by PCOMPS_1:38;
then A18: g3 is_homeomorphism by A14,A15,A16,A17,COMPTS_1:26;
reconsider h=g3*g as map
of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(-1,1)
by TOPMETR:27;
A19: h is_homeomorphism by A10,A18,TOPMETR:27,TOPS_2:71;
A20: 0 in dom g by A11,TOPREAL5:1;
A21: 1 in dom g by A11,TOPREAL5:1;
A22: s1 in [.0,1.] by A10,TOPREAL5:1;
A23: s2 in [.0,1.] by A10,TOPREAL5:1;
A24: -1=(|[-1,0]|)`1 by EUCLID:56.=proj1.(|[-1,0]|) by PSCOMP_1:def 28
.=g3.(|[-1,0]|) by A8,A13,FUNCT_1:72 .= h.1 by A8,A10,A21,FUNCT_1:23
;
A25: 1=(|[1,0]|)`1 by EUCLID:56.=proj1.(|[1,0]|) by PSCOMP_1:def 28
.=g3.(|[1,0]|) by A9,A12,FUNCT_1:72 .= h.0 by A9,A10,A20,FUNCT_1:23;
A26: p1`1=g0.p1 by PSCOMP_1:def 28
.=g3.p1 by A6,FUNCT_1:72 .= h.s1 by A10,A11,A22,FUNCT_1:23;
p2`1=g0.p2 by PSCOMP_1:def 28
.=g3.p2 by A7,FUNCT_1:72 .= h.s2 by A10,A11,A23,FUNCT_1:23;
hence s1 <= s2 by A1,A19,A22,A23,A24,A25,A26,Th12;
end;
then p1 in Lower_Arc(P) & p2 in Lower_Arc(P) & not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A1,A6,A7,JORDAN5C:def 3;
hence LE p1,p2,P by JORDAN6:def 10;
end;
theorem Th60: for cn being Real,q being Point of TOP-REAL 2 st
-1<cn & cn<1 & q`2<=0 holds (for p being Point of TOP-REAL 2 st
p=(cn-FanMorphS).q holds p`2<=0)
proof let cn be Real,q be Point of TOP-REAL 2;
assume A1: -1<cn & cn<1 & q`2<=0;
let p be Point of TOP-REAL 2;
assume A2: p=(cn-FanMorphS).q;
per cases by A1;
suppose A3: q`2<0;
now per cases;
case q`1/|.q.|<cn;
hence p`2<=0 by A1,A2,A3,JGRAPH_4:145;
case q`1/|.q.|>=cn;
hence p`2<=0 by A1,A2,A3,JGRAPH_4:144;
end;
hence p`2<=0;
suppose q`2=0;
hence p`2<=0 by A2,JGRAPH_4:120;
end;
theorem Th61: for cn being Real,p1,p2,q1,q2 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st -1<cn & cn<1 & P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & q1=(cn-FanMorphS).p1 & q2=(cn-FanMorphS).p2
holds LE q1,q2,P
proof let cn be Real,p1,p2,q1,q2 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: -1<cn & cn<1 & P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & q1=(cn-FanMorphS).p1 & q2=(cn-FanMorphS).p2;
then A2: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by Th37;
A3: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A1,Th38;
A4: P is_simple_closed_curve by A1,JGRAPH_3:36;
W-min(P)=|[-1,0]| by A1,Th32;
then A5: (W-min(P))`2=0 by EUCLID:56;
then A6: (cn-FanMorphS).(W-min(P))=W-min(P) by JGRAPH_4:120;
W-min(P) in the carrier of TOP-REAL 2;
then A7: W-min(P) in dom ((cn-FanMorphS)) by FUNCT_2:def 1;
p2 in the carrier of TOP-REAL 2;
then A8: p2 in dom ((cn-FanMorphS)) by FUNCT_2:def 1;
A9: (cn-FanMorphS) is one-to-one by A1,JGRAPH_4:140;
A10: Upper_Arc(P) c= P by A1,Th36;
A11: Lower_Arc(P) c= P by A1,Th36;
A12: now per cases by A1,JORDAN6:def 10;
case p1 in Upper_Arc(P);
hence p1 in P by A10;
case p1 in Lower_Arc(P);
hence p1 in P by A11;
end;
A13: now assume A14: q2=W-min(P);
then p2=W-min(P) by A1,A6,A7,A8,A9,FUNCT_1:def 8;
then LE p2,p1,P by A4,A12,JORDAN7:3;
then A15: q1=q2 by A1,A4,JORDAN6:72;
W-min(P) in Lower_Arc(P) by A4,JORDAN7:1;
then LE q1,q2,P by A4,A11,A14,A15,JORDAN6:71;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P) by JORDAN6:def 10;
end;
per cases by A1,JORDAN6:def 10;
suppose A16: p1 in Upper_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P);
then consider p8 being Point of TOP-REAL 2 such that
A17: p8=p1 & p8 in P & p8`2>=0 by A2;
consider p9 being Point of TOP-REAL 2 such that
A18: p9=p2 & p9 in P & p9`2<=0 by A3,A16;
A19: |.q2.|=|.p2.| by A1,JGRAPH_4:135;
consider p10 being Point of TOP-REAL 2 such that
A20: p10=p2 & |.p10.|=1 by A1,A18;
A21: q2 in P by A1,A19,A20;
q2`2<=0 by A1,A18,Th60;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P) by A1,A3,A13,A16,A17,A21,
JGRAPH_4:120;
suppose A22: p1 in Upper_Arc(P) & p2 in Upper_Arc(P) &
LE p1,p2,Upper_Arc(P),W-min(P),E-max(P);
then consider p8 being Point of TOP-REAL 2 such that
A23: p8=p1 & p8 in P & p8`2>=0 by A2;
consider p9 being Point of TOP-REAL 2 such that
A24: p9=p2 & p9 in P & p9`2>=0 by A2,A22;
p1=(cn-FanMorphS).p1 by A23,JGRAPH_4:120;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P) by A1,A22,A24,JGRAPH_4:120;
suppose A25: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) &
LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) & not p1 in Upper_Arc(P);
A26: |.q1.|=|.p1.| by A1,JGRAPH_4:135;
A27: |.q2.|=|.p2.| by A1,JGRAPH_4:135;
consider p8 being Point of TOP-REAL 2 such that
A28: p8=p1 & p8 in P & p8`2<=0 by A3,A25;
A29: q1`2<=0 by A1,A28,Th60;
consider p9 being Point of TOP-REAL 2 such that
A30: p9=p2 & p9 in P & p9`2<=0 by A3,A25;
A31: q2`2<=0 by A1,A30,Th60;
consider p10 being Point of TOP-REAL 2 such that
A32: p10=p1 & |.p10.|=1 by A1,A28;
consider p11 being Point of TOP-REAL 2 such that
A33: p11=p2 & |.p11.|=1 by A1,A30;
A34: q1 in P by A1,A26,A32;
A35: q2 in P by A1,A27,A33;
now per cases;
case A36: p1=W-min(P); then p1=(cn-FanMorphS).p1 by A5,JGRAPH_4:120;
then LE q1,q2,P by A1,A4,A35,A36,JORDAN7:3;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P) by JORDAN6:def 10;
case A37: p1<>W-min(P);
now per cases by A1,A28,A30,A37,Th51;
case A38: p1`1=p2`1;
(p1`1)^2+(p1`2)^2=1^2 by A32,JGRAPH_3:10
.=(p1`1)^2+(p2`2)^2 by A33,A38,JGRAPH_3:10;
then (p1`2)^2=(p1`1)^2+(p2`2)^2 -(p1`1)^2 by XCMPLX_1:26
.=(p2`2)^2 by XCMPLX_1:26;
then A39: p1`2=p2`2 or p1`2=-p2`2 by JGRAPH_3:1;
A40: p1=|[p1`1,p1`2]| by EUCLID:57;
A41: p2=|[p2`1,p2`2]| by EUCLID:57;
now assume A42: p1`2=-p2`2; then p2`2=0 by A28,A30,REAL_1:66;
hence p1=p2 by A38,A41,A42,EUCLID:57;
end;
then LE q1,q2,P by A1,A4,A35,A38,A39,A40,A41,JORDAN6:71;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P) by JORDAN6:def 10;
case p1`1>p2`1;
then A43: p1`1/|.p1.|>p2`1/|.p2.| by A32,A33;
A44: q2<> W-min(P) by A1,A6,A7,A8,A9,A25,FUNCT_1:def 8;
q1`1/|.q1.|>=q2`1/|.q2.| by A1,A28,A30,A32,A33,A43,Th30;
then LE q1,q2,P by A1,A26,A27,A29,A31,A32,A33,A34,A35,A44,Th59;
hence
q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P) by JORDAN6:def 10;
end;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P);
end;
hence
q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) &
LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) &
LE q1,q2,Lower_Arc(P),E-max(P),W-min(P);
end;
theorem Th62: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`1<0 & p1`2>=0 & p2`1<0 & p2`2>=0
& p3`1<0 & p3`2>=0 & p4`1<0 & p4`2>=0
ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`1<0 & p1`2>=0 & p2`1<0 & p2`2>=0
& p3`1<0 & p3`2>=0 & p4`1<0 & p4`2>=0;
then P is_simple_closed_curve by JGRAPH_3:36;
then A2: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p3 in P & p4 in P &
LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`1<0 & p1`2>=0 & p2`1<0 & p2`2>=0
& p3`1<0 & p3`2>=0 & p4`1<0 & p4`2>=0 by A1,JORDAN7:5;
consider r being real number such that
A3: p4`1<r & r<0 by A1,REAL_1:75;
reconsider r1=r as Real by XREAL_0:def 1;
set s=sqrt(1-r1^2);
(p4`1)^2>r1^2 by A3,Th2;
then A4: 1-(p4`1)^2<1-r1^2 by REAL_1:92;
consider p being Point of TOP-REAL 2 such that
A5: p=p4 & |.p.|=1 by A2;
consider p11 being Point of TOP-REAL 2 such that
A6: p11=p1 & |.p11.|=1 by A2;
consider p22 being Point of TOP-REAL 2 such that
A7: p22=p2 & |.p22.|=1 by A2;
consider p33 being Point of TOP-REAL 2 such that
A8: p33=p3 & |.p33.|=1 by A2;
-1<=p4`1 by A5,Th3;
then A9: -1<=r1 by A3,AXIOMS:22;
r1<=1 by A3,AXIOMS:22;
then r1^2<=1^2 by A9,JGRAPH_2:7;
then A10: 1-r1^2>=0 by SQUARE_1:12,59;
then A11: s^2=1-r1^2 by SQUARE_1:def 4;
then 1-s^2=1-1+r1^2 by XCMPLX_1:37 .=r1^2;
then 1-s^2>0 by A3,SQUARE_1:74;
then 1-s^2+s^2>0+s^2 by REAL_1:67;
then 1>0+s^2 by XCMPLX_1:27;
then A12: -1<s & s<1 by JGRAPH_4:5;
then consider f1 being map of TOP-REAL 2,TOP-REAL 2 such that
A13: f1=s-FanMorphW & f1 is_homeomorphism by JGRAPH_4:48;
A14: for q being Point of TOP-REAL 2 holds |.(f1.q).|=|.q.|
by A13,JGRAPH_4:40;
A15: s>=0 by A10,SQUARE_1:def 4;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
A16: |.q11.|=1 by A6,A13,JGRAPH_4:40;
then A17: q11 in P by A1;
A18: |.q22.|=1 by A7,A13,JGRAPH_4:40;
then A19: q22 in P by A1;
A20: |.q33.|=1 by A8,A13,JGRAPH_4:40;
then A21: q33 in P by A1;
A22: |.q44.|=1 by A5,A13,JGRAPH_4:40;
then A23: q44 in P by A1;
A24: -(p4`1)>0 by A1,REAL_1:66;
A25: p1`1<p2`1 or p1=p2 by A1,Th49;
A26: p2`1<p3`1 or p2=p3 by A1,Th49;
A27: p3`1<p4`1 or p3=p4 by A1,Th49;
then A28: p2`1<=p4`1 by A26,AXIOMS:22;
then p1`1<=p4`1 by A25,AXIOMS:22;
then -(p1`1)>= -(p4`1) by REAL_1:50;
then (-(p1`1))^2>=(-(p4`1))^2 by A24,SQUARE_1:77;
then (-(p1`1))^2>=((p4`1))^2 by SQUARE_1:61;
then ((p1`1))^2>=((p4`1))^2 by SQUARE_1:61;
then 1-((p1`1))^2<=1-((p4`1))^2 by REAL_2:106;
then A29: 1-(p1`1)^2< s^2 by A4,A11,AXIOMS:22;
1^2=(p1`1)^2+(p1`2)^2 by A6,JGRAPH_3:10;
then 1-((p1`1))^2=((p1`2))^2 by SQUARE_1:59,XCMPLX_1:26;
then A30: p1`2/|.p1.|<s by A6,A15,A29,JGRAPH_2:6;
then A31: q11`1<0 & q11`2<0 by A1,A12,A13,JGRAPH_4:50;
-(p2`1)>= -(p4`1) by A28,REAL_1:50;
then (-(p2`1))^2>=(-(p4`1))^2 by A24,SQUARE_1:77;
then (-(p2`1))^2>=((p4`1))^2 by SQUARE_1:61;
then ((p2`1))^2>=((p4`1))^2 by SQUARE_1:61;
then 1-((p2`1))^2<=1-((p4`1))^2 by REAL_2:106;
then A32: 1-(p2`1)^2< s^2 by A4,A11,AXIOMS:22;
1^2=(p2`1)^2+(p2`2)^2 by A7,JGRAPH_3:10;
then 1-((p2`1))^2=((p2`2))^2 by SQUARE_1:59,XCMPLX_1:26;
then A33: p2`2/|.p2.|<s by A7,A15,A32,JGRAPH_2:6;
then A34: q22`1<0 & q22`2<0 by A1,A12,A13,JGRAPH_4:50;
-(p3`1)>= -(p4`1) by A27,REAL_1:50;
then (-(p3`1))^2>=(-(p4`1))^2 by A24,SQUARE_1:77;
then (-(p3`1))^2>=((p4`1))^2 by SQUARE_1:61;
then ((p3`1))^2>=((p4`1))^2 by SQUARE_1:61;
then 1-((p3`1))^2<=1-((p4`1))^2 by REAL_2:106;
then A35: 1-(p3`1)^2< s^2 by A4,A11,AXIOMS:22;
1^2=(p3`1)^2+(p3`2)^2 by A8,JGRAPH_3:10;
then 1-((p3`1))^2=((p3`2))^2 by SQUARE_1:59,XCMPLX_1:26;
then A36: p3`2/|.p3.|<s by A8,A15,A35,JGRAPH_2:6;
then A37: q33`1<0 & q33`2<0 by A1,A12,A13,JGRAPH_4:50;
1^2=(p4`1)^2+(p4`2)^2 by A5,JGRAPH_3:10;
then 1-((p4`1))^2=((p4`2))^2 by SQUARE_1:59,XCMPLX_1:26;
then p4`2/|.p4.|<s by A4,A5,A11,A15,JGRAPH_2:6;
then A38: (q11`1<0 & q11`2<0)&(q22`1<0 & q22`2<0)&(q33`1<0 & q33`2<0)
&(q44`1<0 & q44`2<0) by A1,A12,A13,A30,A33,A36,JGRAPH_4:50;
p1`2/|.p1.|<p2`2/|.p2.| or p1=p2 by A1,A6,A7,Th49;
then q11`2/|.q11.|<q22`2/|.q22.| or p1=p2 by A1,A12,A13,JGRAPH_4:53;
then A39: LE q11,q22,P by A1,A16,A17,A18,A19,A31,A34,Th54;
p2`2/|.p2.|<p3`2/|.p3.| or p2=p3 by A1,A7,A8,Th49;
then q22`2/|.q22.|<q33`2/|.q33.| or p2=p3 by A1,A12,A13,JGRAPH_4:53;
then A40: LE q22,q33,P by A1,A18,A19,A20,A21,A34,A37,Th54;
p3`2/|.p3.|<p4`2/|.p4.| or p3=p4 by A1,A5,A8,Th49;
then q33`2/|.q33.|<q44`2/|.q44.| or p3=p4 by A1,A12,A13,JGRAPH_4:53;
then LE q33,q44,P by A1,A20,A21,A22,A23,A38,Th54;
hence thesis by A13,A14,A38,A39,A40;
end;
theorem Th63: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`2>=0 & p2`2>=0
& p3`2>=0 & p4`2>0
ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2>=0)&(q2`1<0 & q2`2>=0)&
(q3`1<0 & q3`2>=0)&(q4`1<0 & q4`2>=0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`2>=0 & p2`2>=0
& p3`2>=0 & p4`2>0;
then P is_simple_closed_curve by JGRAPH_3:36;
then A2: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p3 in P & p4 in P &
LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`2>=0 & p2`2>=0
& p3`2>=0 & p4`2>0 by A1,JORDAN7:5;
then consider p being Point of TOP-REAL 2 such that
A3: p=p4 & |.p.|=1;
consider p11 being Point of TOP-REAL 2 such that
A4: p11=p1 & |.p11.|=1 by A2;
consider p22 being Point of TOP-REAL 2 such that
A5: p22=p2 & |.p22.|=1 by A2;
consider p33 being Point of TOP-REAL 2 such that
A6: p33=p3 & |.p33.|=1 by A2;
A7: -1<=p4`1 & p4`1<=1 by A3,Th3;
now assume p4`1=1;
then 1=1+(p4`2)^2 by A3,JGRAPH_3:10,SQUARE_1:59;
then 1-1=(p4`2)^2 by XCMPLX_1:26;
hence contradiction by A1,SQUARE_1:73;
end;
then p4`1<1 by A7,REAL_1:def 5;
then consider r being real number such that
A8: p4`1<r & r<1 by REAL_1:75;
reconsider r1=r as Real by XREAL_0:def 1;
A9: -1<r1 & r1<1 by A7,A8,AXIOMS:22;
then consider f1 being map of TOP-REAL 2,TOP-REAL 2 such that
A10: f1=r1-FanMorphN & f1 is_homeomorphism by JGRAPH_4:81;
A11: for q being Point of TOP-REAL 2 holds |.(f1.q).|=|.q.|
by A10,JGRAPH_4:73;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
A12: |.q11.|=1 by A4,A10,JGRAPH_4:73;
then A13: q11 in P by A1;
A14: |.q22.|=1 by A5,A10,JGRAPH_4:73;
then A15: q22 in P by A1;
A16: |.q33.|=1 by A6,A10,JGRAPH_4:73;
then A17: q33 in P by A1;
A18: |.q44.|=1 by A3,A10,JGRAPH_4:73;
then A19: q44 in P by A1;
A20: p1`1<p2`1 or p1=p2 by A1,Th50;
A21: p2`1<p3`1 or p2=p3 by A1,Th50;
A22: p3`1<p4`1 or p3=p4 by A1,Th50;
then A23: p3`1<r1 by A8,AXIOMS:22;
then A24: p2`1<r1 by A21,AXIOMS:22;
A25: p2`1/|.p2.|<r1 by A5,A21,A23,AXIOMS:22;
then A26: q22`2>=0 & q22`1<0 by A1,A5,A9,A10,Th23;
A27: p1`1/|.p1.|<r1 by A4,A20,A24,AXIOMS:22;
then A28: q11`2>=0 & q11`1<0 by A1,A4,A9,A10,Th23;
p4`1/|.p4.|<r1 by A3,A8;
then A29: q44`1<0 & q44`2>0 by A1,A9,A10,JGRAPH_4:83;
A30: (q11`1<0 & q11`2>=0 or q11`1<0 & q11`2=0)&(q22`1<0 & q22`2>=0)
by A1,A4,A5,A9,A10,A25,A27,Th23;
p1`1/|.p1.|<p2`1/|.p2.| or p1=p2 by A1,A4,A5,Th50;
then q11`1/|.q11.|<q22`1/|.q22.| or p1=p2
by A1,A4,A5,A9,A10,Th24;
then A31: LE q11,q22,P by A1,A12,A13,A14,A15,A26,A28,Th56;
p3`1/|.p3.|<r1 by A6,A8,A22,AXIOMS:22;
then A32: q33`2>=0 & q33`1<0 by A1,A6,A9,A10,Th23;
p2`1/|.p2.|<p3`1/|.p3.| or p2=p3 by A1,A5,A6,Th50;
then q22`1/|.q22.|<q33`1/|.q33.| or p2=p3
by A1,A5,A6,A9,A10,Th24;
then A33: LE q22,q33,P by A1,A14,A15,A16,A17,A26,A32,Th56;
p3`1/|.p3.|<p4`1/|.p4.| or p3=p4 by A1,A3,A6,Th50;
then q33`1/|.q33.|<q44`1/|.q44.| or p3=p4
by A1,A3,A6,A9,A10,Th24;
then LE q33,q44,P by A1,A16,A17,A18,A19,A29,A32,Th56;
hence thesis by A10,A11,A29,A30,A31,A32,A33;
end;
theorem Th64: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`2>=0 & p2`2>=0
& p3`2>=0 & p4`2>0
ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& p1`2>=0 & p2`2>=0
& p3`2>=0 & p4`2>0;
then consider f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 such that
A2: f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2>=0)&(q2`1<0 & q2`2>=0)&
(q3`1<0 & q3`2>=0)&(q4`1<0 & q4`2>=0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by Th63;
consider f2 being map of TOP-REAL 2,TOP-REAL 2,
q1b,q2b,q3b,q4b being Point of TOP-REAL 2 such that
A3: f2 is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.|)&
q1b=f2.q1 & q2b=f2.q2 & q3b=f2.q3 & q4b=f2.q4 &
(q1b`1<0 & q1b`2<0)&(q2b`1<0 & q2b`2<0)&
(q3b`1<0 & q3b`2<0)&(q4b`1<0 & q4b`2<0)&
LE q1b,q2b,P & LE q2b,q3b,P & LE q3b,q4b,P by A1,A2,Th62;
reconsider f3=f2*f as map of TOP-REAL 2,TOP-REAL 2;
A4: f3 is_homeomorphism by A2,A3,TOPS_2:71;
A5: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof let q be Point of TOP-REAL 2;
dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then f3.q=f2.(f.q) by FUNCT_1:23;
hence |.f3.q.|=|.(f.q).| by A3 .=|.q.| by A2;
end;
A6: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then A7: f3.p1=q1b by A2,A3,FUNCT_1:23;
A8: f3.p2=q2b by A2,A3,A6,FUNCT_1:23;
A9: f3.p3=q3b by A2,A3,A6,FUNCT_1:23;
f3.p4=q4b by A2,A3,A6,FUNCT_1:23;
hence thesis by A3,A4,A5,A7,A8,A9;
end;
theorem Th65: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& (p1`2>=0 or p1`1>=0)& (p2`2>=0 or p2`1>=0)
& (p3`2>=0 or p3`1>=0)& (p4`2>0 or p4`1>0)
ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
q1`2>=0 & q2`2>=0 &
q3`2>=0 & q4`2>0 &
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& (p1`2>=0 or p1`1>=0)& (p2`2>=0 or p2`1>=0)
& (p3`2>=0 or p3`1>=0)& (p4`2>0 or p4`1>0);
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p3 in P & p4 in P &
LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& (p1`2>=0 or p1`1>=0)& (p2`2>=0 or p2`1>=0)
& (p3`2>=0 or p3`1>=0)& (p4`2>0 or p4`1>0) by A1,JORDAN7:5;
then consider p44 being Point of TOP-REAL 2 such that
A4: p44=p4 & |.p44.|=1;
consider p11 being Point of TOP-REAL 2 such that
A5: p11=p1 & |.p11.|=1 by A3;
consider p22 being Point of TOP-REAL 2 such that
A6: p22=p2 & |.p22.|=1 by A3;
consider p33 being Point of TOP-REAL 2 such that
A7: p33=p3 & |.p33.|=1 by A3;
A8: -1<=p4`2 & p4`2<=1 by A4,Th3;
now assume A9: p4`2=-1;
1=(p4`1)^2+(p4`2)^2 by A4,JGRAPH_3:10,SQUARE_1:59
.=(p4`1)^2+1 by A9,SQUARE_1:59,61;
then 1-1=(p4`1)^2 by XCMPLX_1:26;
hence contradiction by A1,A9,SQUARE_1:73;
end;
then p4`2> -1 by A8,REAL_1:def 5;
then consider r being real number such that
A10: -1<r & r<p4`2 by REAL_1:75;
reconsider r1=r as Real by XREAL_0:def 1;
A11: -1<r1 & r1<1 by A8,A10,AXIOMS:22;
then consider f1 being map of TOP-REAL 2,TOP-REAL 2 such that
A12: f1=r1-FanMorphE & f1 is_homeomorphism by JGRAPH_4:112;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
A13: |.q11.|=1 by A5,A12,JGRAPH_4:104;
then A14: q11 in P by A1;
A15: |.q22.|=1 by A6,A12,JGRAPH_4:104;
then A16: q22 in P by A1;
A17: |.q33.|=1 by A7,A12,JGRAPH_4:104;
then A18: q33 in P by A1;
A19: |.q44.|=1 by A4,A12,JGRAPH_4:104;
then A20: q44 in P by A1;
now per cases;
case A21: p4`2<=0;
A22: p4`2/|.p4.|>r1 by A4,A10;
then A23: q44`1>0 & q44`2>=0 by A1,A11,A12,A21,JGRAPH_4:113;
A24: now assume A25: q44`2=0;
1^2=(q44`1)^2+(q44`2)^2 by A19,JGRAPH_3:10
.=(q44`1)^2 by A25,SQUARE_1:60;
then q44`1=-1 or q44`1=1 by JGRAPH_3:2,SQUARE_1:59;
then A26: q44=|[1,0]| by A1,A11,A12,A21,A22,A25,EUCLID:57,JGRAPH_4:113;
set q8=|[sqrt(1-r1^2),r1]|;
A27: q8`1=sqrt(1-r1^2) & q8`2=r1 by EUCLID:56;
then A28: |.q8.|=sqrt((sqrt(1-r1^2))^2+r1^2)by JGRAPH_3:10;
1^2>r1^2 by A11,JGRAPH_2:8;
then A29: 1-r1^2>0 by SQUARE_1:11,59;
then A30: |.q8.|=sqrt((1-r1^2)+r1^2) by A28,SQUARE_1:def 4
.=1 by SQUARE_1:83,XCMPLX_1:27;
then A31: q8`2/|.q8.|=r1 by EUCLID:56;
A32: q8`1>0 by A27,A29,SQUARE_1:93;
set r8=f1.q8;
A33: r8`1>0 & r8`2=0 by A11,A12,A31,A32,JGRAPH_4:118;
|.r8.|=1 by A12,A30,JGRAPH_4:104;
then 1^2=(r8`1)^2+(r8`2)^2 by JGRAPH_3:10
.=(r8`1)^2 by A33,SQUARE_1:60;
then r8`1=-1 or r8`1=1 by JGRAPH_3:2,SQUARE_1:59;
then A34: f1.(|[sqrt(1-r1^2),r1]|)=|[1,0]| by A33,EUCLID:57;
A35: f1 is one-to-one by A11,A12,JGRAPH_4:109;
dom f1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then p4=|[sqrt(1-r1^2),r1]| by A26,A34,A35,FUNCT_1:def 8;
hence contradiction by A10,EUCLID:56;
end;
A36: Lower_Arc(P)
={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2<=0}
by A1,Th38;
A37: Upper_Arc(P)
={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0}
by A1,Th37;
A38: now per cases;
case A39: p3`1<=0;
then A40: q33=p3 by A12,JGRAPH_4:89;
A41: now per cases by A39;
case p3`1=0;
then 1=0+(q33`2)^2 by A7,A40,JGRAPH_3:10,SQUARE_1:59,60
.=(q33`2)^2;
then A42: q33`2=-1 or q33`2=1 by JGRAPH_3:2;
now assume q33`2=-1;
then -1>=p4`2 by A1,A21,A40,Th53;
hence contradiction by A10,AXIOMS:22;
end;
hence q33`2>=0 by A42;
case p3`1<0;
hence q33`2>=0 by A1,A12,JGRAPH_4:89;
end;
A43: q33`1<q44`1 by A1,A11,A12,A21,A22,A39,A40,JGRAPH_4:113;
now per cases;
case A44: p2<> W-min(P);
A45: now assume A46: p2`2<0;
then A47: p2 in Lower_Arc(P) by A3,A36;
p3 in Upper_Arc(P) by A3,A37,A40,A41;
then LE p3,p2,P by A44,A47,JORDAN6:def 10;
hence contradiction by A1,A2,A40,A41,A46,JORDAN6:72;
end;
then A48: p2`1<=p3`1 by A1,A40,A41,Th50;
then A49: q22=p2 by A12,A39,JGRAPH_4:89;
now per cases;
case A50: p1<> W-min(P);
A51: now assume A52: p1`2<0;
then A53: p1 in Lower_Arc(P) by A3,A36;
p2 in Upper_Arc(P) by A3,A37,A45;
then LE p2,p1,P by A50,A53,JORDAN6:def 10;
hence contradiction by A1,A2,A45,A52,JORDAN6:72;
end;
then p1`1<=p2`1 by A1,A45,Th50;
hence q11`2>=0 & q22`2>=0 &
q33`2>=0 & q44`2>0 &
LE q11,q22,P & LE q22,q33,P & LE q33,q44,P
by A1,A12,A18,A20,A23,A24,A39,A41,A43,A45,A48,A49,A51,Th57,JGRAPH_4:89
;
case A54: p1=W-min(P);
A55: W-min(P)=|[-1,0]| by A1,Th32;
then p1`1=-1 & p1`2=0 by A54,EUCLID:56;
then p1=q11 by A12,JGRAPH_4:89;
hence q11`2>=0 & q22`2>=0 &
q33`2>=0 & q44`2>0 &
LE q11,q22,P & LE q22,q33,P & LE q33,q44,P
by A3,A12,A20,A23,A24,A39,A40,A41,A43,A45,A48,A54,A55,Th57,EUCLID:56,
JGRAPH_4:89;
end;
hence q11`2>=0 & q22`2>=0 &
q33`2>=0 & q44`2>0 &
LE q11,q22,P & LE q22,q33,P & LE q33,q44,P;
case A56: p2=W-min(P);
A57: W-min(P)=|[-1,0]| by A1,Th32;
then A58: p2`1=-1 & p2`2=0 by A56,EUCLID:56;
then A59: p2=q22 by A12,JGRAPH_4:89;
A60: now assume A61: p1`2<0;
then A62: p1 in Lower_Arc(P) by A3,A36;
A63: p2 in Upper_Arc(P) by A16,A37,A58,A59;
A64: p1<>W-min(P) by A57,A61,EUCLID:56;
LE p2,p1,P by A56,A58,A61,A62,A63,JORDAN6:def 10;
hence contradiction by A1,A2,A56,A64,JORDAN6:72;
end;
then p1`1<=p2`1 by A1,A58,Th50;
then p1`1<=0 by A58,AXIOMS:22;
hence q11`2>=0 & q22`2>=0 &
q33`2>=0 & q44`2>0 &
LE q11,q22,P & LE q22,q33,P & LE q33,q44,P
by A1,A12,A18,A20,A23,A24,A41,A43,A56,A57,A59,A60,Th57,EUCLID:56,
JGRAPH_4:89;
end;
hence q11`2>=0 & q22`2>=0 & q33`2>=0 & q44`2>0 &
LE q11,q22,P & LE q22,q33,P & LE q33,q44,P;
case A65: p3`1>0;
A66: now per cases;
case A67: p3<>p4;
then p3`2>p4`2 by A1,A21,A65,Th53;
then A68: p3`2/|.p3.|>=r1 by A7,A10,AXIOMS:22;
then A69: q33`1>0 & q33`2>=0 by A11,A12,A65,JGRAPH_4:113;
p3`2/|.p3.|>p4`2/|.p4.| by A1,A4,A7,A21,A65,A67,Th53;
then A70: q33`2/|.q33.|>q44`2/|.q44.| by A1,A4,A7,A11,A12,A21,A65,Th27;
1^2=(q33`1)^2+(q33`2)^2 by A17,JGRAPH_3:10;
then A71: 1^2-(q33`2)^2=(q33`1)^2 by XCMPLX_1:26;
then A72: (q33`1)=sqrt(1^2-((q33`2))^2) by A69,SQUARE_1:89;
1^2=(q44`1)^2+(q44`2)^2 by A19,JGRAPH_3:10;
then 1^2-(q44`2)^2=(q44`1)^2 by XCMPLX_1:26;
then A73: (q44`1)=sqrt(1^2-((q44`2))^2) by A23,SQUARE_1:89;
((q33`2))^2 > ((q44`2))^2 by A17,A19,A23,A70,SQUARE_1:78;
then A74: 1^2- ((q33`2))^2 < 1^2-((q44`2))^2 by REAL_1:92;
1^2-((q33`2))^2>=0 by A71,SQUARE_1:72;
then A75: q33`1< q44`1 by A72,A73,A74,SQUARE_1:95;
A76: now assume p2`1=0;
then 1^2=0+(p2`2)^2 by A6,JGRAPH_3:10,SQUARE_1:60;
hence p2`2=1 or p2`2=-1 by JGRAPH_3:1;
end;
A77: now assume A78: p2`1=0 & p2`2=-1;
then p2`2<=p4`2 by A4,Th3;
then A79: LE p4,p2,P by A3,A21,A78,Th58;
LE p2,p4,P by A1,A2,JORDAN6:73;
then p2=p4 by A2,A79,JORDAN6:72;
hence contradiction by A1,A78;
end;
now per cases by A1,A76,A77;
case A80: p2`1<=0 & p2`2>=0;
then A81: q22=p2 by A12,JGRAPH_4:89;
A82: q33`2>=0 by A11,A12,A65,A68,JGRAPH_4:113;
q22`1<=q33`1 by A69,A80,A81,AXIOMS:22;
hence q22`2>=0 & LE q22,q33,P by A3,A18,A80,A81,A82,Th57;
case A83: p2`1>0;
then A84: q22`1>0 by A11,A12,Th25;
now per cases;
case p2=p3;
hence q22`2>=0 & LE q22,q33,P by A2,A11,A12,A18,A65,A68,JGRAPH_4:113
,JORDAN6:71;
case p2<>p3;
then p2`2/|.p2.|>p3`2/|.p3.| by A1,A6,A7,A65,A83,Th53;
then q22`2/|.q22.|>q33`2/|.q33.|
by A6,A7,A11,A12,A65,A83,Th27;
hence q22`2>=0 & LE q22,q33,P
by A1,A15,A16,A17,A18,A69,A84,Th58,AXIOMS:22;
end;
hence q22`2>=0 & LE q22,q33,P;
end;
hence q22`2>=0 & LE q22,q33,P & q33`2>=0 & LE q33,q44,P
by A1,A18,A20,A23,A69,A75,Th57;
case A85: p3=p4;
then A86: p3`2/|.p3.|>=r1 by A7,A10;
then A87: q33`1>0 & q33`2>=0 by A11,A12,A65,JGRAPH_4:113;
A88: now assume p2`1=0;
then 1^2=0+(p2`2)^2 by A6,JGRAPH_3:10,SQUARE_1:60;
hence p2`2=1 or p2`2=-1 by JGRAPH_3:1;
end;
A89: now assume A90: p2`1=0 & p2`2=-1;
then A91: LE p4,p2,P by A3,A8,A21,Th58;
LE p2,p4,P by A1,A2,JORDAN6:73;
then p2=p4 by A2,A91,JORDAN6:72;
hence contradiction by A1,A90;
end;
now per cases by A1,A88,A89;
case A92: p2`1<=0 & p2`2>=0;
then A93: q22=p2 by A12,JGRAPH_4:89;
A94: q33`2>=0 by A11,A12,A65,A86,JGRAPH_4:113;
q22`1<=q33`1 by A87,A92,A93,AXIOMS:22;
hence q22`2>=0 & LE q22,q33,P by A3,A18,A92,A93,A94,Th57;
case A95: p2`1>0;
then A96: q22`1>0 by A11,A12,Th25;
now per cases;
case p2=p3;
hence q22`2>=0 & LE q22,q33,P by A2,A11,A12,A18,A65,A86,JGRAPH_4:113
,JORDAN6:71;
case p2<>p3;
then p2`2/|.p2.|>p3`2/|.p3.| by A1,A6,A7,A65,A95,Th53;
then q22`2/|.q22.|>q33`2/|.q33.|
by A6,A7,A11,A12,A65,A95,Th27;
hence q22`2>=0 & LE q22,q33,P
by A1,A15,A16,A17,A18,A87,A96,Th58,AXIOMS:22;
end;
hence q22`2>=0 & LE q22,q33,P;
end;
hence q22`2>=0 & LE q22,q33,P & q33`2>=0 & LE q33,q44,P
by A1,A18,A23,A85,Th57;
end;
A97: now assume p1`1=0;
then 1^2=0+(p1`2)^2 by A5,JGRAPH_3:10,SQUARE_1:60;
hence p1`2=1 or p1`2=-1 by JGRAPH_3:1;
end;
A98: now assume A99: p1`1=0 & p1`2=-1;
then A100: LE p4,p1,P by A3,A8,A21,Th58;
LE p1,p3,P by A1,A2,JORDAN6:73;
then LE p1,p4,P by A1,A2,JORDAN6:73;
then p1=p4 by A2,A100,JORDAN6:72;
hence contradiction by A1,A99;
end;
A101: now assume p2`1=0;
then 1^2=0+(p2`2)^2 by A6,JGRAPH_3:10,SQUARE_1:60;
hence p2`2=1 or p2`2=-1 by JGRAPH_3:1;
end;
A102: now assume A103: p2`1=0 & p2`2=-1;
then p2`2<=p4`2 by A4,Th3;
then A104: LE p4,p2,P by A3,A21,A103,Th58;
LE p2,p4,P by A1,A2,JORDAN6:73;
then p2=p4 by A2,A104,JORDAN6:72;
hence contradiction by A1,A103;
end;
now per cases by A1,A97,A98;
case A105: p1`1<=0 & p1`2>=0;
then A106: p1=q11 by A12,JGRAPH_4:89;
A107: q11`2>=0 by A12,A105,JGRAPH_4:89;
now per cases by A1,A101,A102;
case p2`1<=0 & p2`2>=0;
hence q11`2>=0 & LE q11,q22,P by A1,A12,A105,A106,JGRAPH_4:89;
case p2`1>0;
then q11`1<q22`1 by A11,A12,A105,A106,Th25;
hence q11`2>=0 & LE q11,q22,P by A1,A14,A16,A66,A107,Th57;
end;
hence q11`2>=0 & LE q11,q22,P;
case A108: p1`1>0;
then A109: q11`1>0 by A11,A12,Th25;
now per cases by A1,A101,A102;
case A110: p2`1<=0 & p2`2>=0;
then A111: p2`1<p1`1 by A108;
now assume A112: p1`2<0;
then A113: p1 in Lower_Arc(P) by A3,A36;
A114: p2 in Upper_Arc(P) by A3,A37,A110;
W-min(P)=|[-1,0]| by A1,Th32;
then p1<>W-min(P) by A112,EUCLID:56;
then LE p2,p1,P by A113,A114,JORDAN6:def 10;
hence contradiction by A1,A2,A108,A110,JORDAN6:72;
end;
then LE p2,p1,P by A3,A110,A111,Th57;
then q11=q22 by A1,A2,JORDAN6:72;
hence q11`2>=0 & LE q11,q22,P by A2,A12,A14,A110,JGRAPH_4:89,JORDAN6
:71;
case A115: p2`1>0;
then A116: q22`1>0 by A11,A12,Th25;
now per cases;
case p1=p2;
hence q11`2>=0 & LE q11,q22,P by A2,A14,A66,JORDAN6:71;
case p1<>p2;
then p1`2/|.p1.|>p2`2/|.p2.| by A1,A5,A6,A108,A115,Th53;
then q11`2/|.q11.|>q22`2/|.q22.|
by A5,A6,A11,A12,A108,A115,Th27;
hence q11`2>=0 & LE q11,q22,P
by A1,A13,A14,A15,A16,A66,A109,A116,Th58,AXIOMS:22;
end;
hence q11`2>=0 & LE q11,q22,P;
end;
hence q11`2>=0 & LE q11,q22,P;
end;
hence q11`2>=0 & q22`2>=0 &
q33`2>=0 & q44`2>0 &
LE q11,q22,P & LE q22,q33,P & LE q33,q44,P by A1,A11,A12,A21,A22,A24,A66
,JGRAPH_4:113;
end;
for q being Point of TOP-REAL 2 holds |.(f1.q).|=|.q.|
by A12,JGRAPH_4:104;
hence ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
q1`2>=0 & q2`2>=0 &
q3`2>=0 & q4`2>0 &
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by A12,A38;
case A117: p4`2>0;
A118: p3 in Upper_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) or
p3 in Upper_Arc(P) & p4 in Upper_Arc(P) &
LE p3,p4,Upper_Arc(P),W-min(P),E-max(P) or
p3 in Lower_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) &
LE p3,p4,Lower_Arc(P),E-max(P),W-min(P) by A1,JORDAN6:def 10;
A119: Upper_Arc(P)={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0}
by A1,Th37;
A120: Lower_Arc(P)={p8 where p8 is Point of TOP-REAL 2:p8 in P & p8`2<=0}
by A1,Th38;
A121: now assume p4 in Lower_Arc(P);
then consider p9 being Point of TOP-REAL 2 such that
A122: p9=p4 & p9 in P & p9`2<=0 by A120;
thus contradiction by A117,A122;
end;
then consider p33 being Point of TOP-REAL 2 such that
A123: p33=p3 & p33 in P & p33`2>=0 by A118,A119;
A124: LE p2,p4,P by A1,A2,JORDAN6:73;
then p2 in Upper_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) or
p2 in Upper_Arc(P) & p4 in Upper_Arc(P) &
LE p2,p4,Upper_Arc(P),W-min(P),E-max(P) or
p2 in Lower_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) &
LE p2,p4,Lower_Arc(P),E-max(P),W-min(P)
by JORDAN6:def 10;
then consider p22 being Point of TOP-REAL 2 such that
A125: p22=p2 & p22 in P & p22`2>=0 by A119,A121;
LE p1,p4,P by A1,A2,A124,JORDAN6:73;
then p1 in Upper_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) or
p1 in Upper_Arc(P) & p4 in Upper_Arc(P) &
LE p1,p4,Upper_Arc(P),W-min(P),E-max(P) or
p1 in Lower_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) &
LE p1,p4,Lower_Arc(P),E-max(P),W-min(P)
by JORDAN6:def 10;
then consider p11 being Point of TOP-REAL 2 such that
A126: p11=p1 & p11 in P & p11`2>=0 by A119,A121;
set f4=id (TOP-REAL 2);
A127: for q being Point of TOP-REAL 2 holds f4.q=q
proof let q be Point of TOP-REAL 2;
f4=id (the carrier of TOP-REAL 2) by GRCAT_1:def 11;
hence f4.q=q by FUNCT_1:35;
end;
then A128: f4.p1=p1 & f4.p2=p2 & f4.p3=p3 & f4.p4=p4;
(for q being Point of TOP-REAL 2 holds |.(f4.q).|=|.q.|) by A127;
hence ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
q1`2>=0 & q2`2>=0 &
q3`2>=0 & q4`2>0 &
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by A1,A117,A123,A125,A126,A128;
end;
hence thesis;
end;
theorem Th66: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& (p1`2>=0 or p1`1>=0)& (p2`2>=0 or p2`1>=0)
& (p3`2>=0 or p3`1>=0)& (p4`2>0 or p4`1>0)
ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
& (p1`2>=0 or p1`1>=0)& (p2`2>=0 or p2`1>=0)
& (p3`2>=0 or p3`1>=0)& (p4`2>0 or p4`1>0);
then consider f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 such that
A2: f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
q1`2>=0 & q2`2>=0 &
q3`2>=0 & q4`2>0 &
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by Th65;
consider f2 being map of TOP-REAL 2,TOP-REAL 2,
q1b,q2b,q3b,q4b being Point of TOP-REAL 2 such that
A3: f2 is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.|)&
q1b=f2.q1 & q2b=f2.q2 & q3b=f2.q3 & q4b=f2.q4 &
(q1b`1<0 & q1b`2<0)&(q2b`1<0 & q2b`2<0)&
(q3b`1<0 & q3b`2<0)&(q4b`1<0 & q4b`2<0)&
LE q1b,q2b,P & LE q2b,q3b,P & LE q3b,q4b,P by A1,A2,Th64;
reconsider f3=f2*f as map of TOP-REAL 2,TOP-REAL 2;
A4: f3 is_homeomorphism by A2,A3,TOPS_2:71;
A5: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof let q be Point of TOP-REAL 2;
dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then f3.q=f2.(f.q) by FUNCT_1:23;
hence |.f3.q.|=|.(f.q).| by A3 .=|.q.| by A2;
end;
A6: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then A7: f3.p1=q1b by A2,A3,FUNCT_1:23;
A8: f3.p2=q2b by A2,A3,A6,FUNCT_1:23;
A9: f3.p3=q3b by A2,A3,A6,FUNCT_1:23;
f3.p4=q4b by A2,A3,A6,FUNCT_1:23;
hence thesis by A3,A4,A5,A7,A8,A9;
end;
theorem for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p4=W-min(P) & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p4=W-min(P) &
LE p1,p2,P & LE p2,p3,P & LE p3,p4,P;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
A4: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p3 in P & p4 in P
& p4=W-min(P) &
LE p1,p2,P & LE p2,p3,P & LE p3,p4,P by A1,A2,JORDAN7:5;
A5: Upper_Arc(P)
={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0}
by A1,Th37;
W-min(P)=|[-1,0]| by A1,Th32;
then A6: (W-min(P))`1=-1 & (W-min(P))`2=0 by EUCLID:56;
then A7: p4 in Upper_Arc(P) by A4,A5;
then A8: p3 in Upper_Arc(P) by A1,Th47;
then LE p4,p3,Upper_Arc(P),W-min(P),E-max(P) by A1,A3,JORDAN5C:10;
then LE p4,p3,P by A7,A8,JORDAN6:def 10;
then A9: p3=p4 by A1,A2,JORDAN6:72;
A10: p2 in Upper_Arc(P) by A1,A8,Th47;
then LE p4,p2,Upper_Arc(P),W-min(P),E-max(P) by A1,A3,JORDAN5C:10;
then A11: LE p4,p2,P by A7,A10,JORDAN6:def 10;
LE p2,p4,P by A1,A2,JORDAN6:73;
then A12: p2=p4 by A2,A11,JORDAN6:72;
A13: p1 in Upper_Arc(P) by A1,A10,Th47;
then LE p4,p1,Upper_Arc(P),W-min(P),E-max(P) by A1,A3,JORDAN5C:10;
then LE p4,p1,P by A7,A13,JORDAN6:def 10;
then p1=p4 by A1,A2,A12,JORDAN6:72;
hence thesis by A1,A6,A9,A12,Th62;
end;
theorem Th68: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P;
then A2: P is_simple_closed_curve by JGRAPH_3:36;
then A3: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
A4: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p3 in P & p4 in P &
LE p1,p2,P & LE p2,p3,P & LE p3,p4,P by A1,A2,JORDAN7:5;
then consider p44 being Point of TOP-REAL 2 such that
A5: p44=p4 & |.p44.|=1;
A6: -1<=p4`1 & p4`1<=1 by A5,Th3;
A7: Lower_Arc(P)
={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2<=0} by A1,Th38;
A8: Upper_Arc(P)
={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0}
by A1,Th37;
A9: W-min(P)=|[-1,0]| by A1,Th32;
then A10: (W-min(P))`1=-1 & (W-min(P))`2=0 by EUCLID:56;
now per cases;
case A11: p4`1=-1;
1=(p4`1)^2+(p4`2)^2 by A5,JGRAPH_3:10,SQUARE_1:59
.=(p4`2)^2+1 by A11,SQUARE_1:59,61;
then 1-1=(p4`2)^2 by XCMPLX_1:26;
then A12: p4`2=0 by SQUARE_1:73;
then A13: p4 in Upper_Arc(P) by A4,A8;
A14: p4=W-min(P) by A9,A11,A12,EUCLID:57;
A15: now per cases;
case A16: p1 in Upper_Arc(P);
then LE p4,p1,Upper_Arc(P),W-min(P),E-max(P)
by A3,A14,JORDAN5C:10;
hence LE p4,p1,P by A13,A16,JORDAN6:def 10;
case not p1 in Upper_Arc(P);
then A17: p1`2<0 by A4,A8;
then p1 in Lower_Arc(P) by A4,A7;
hence LE p4,p1,P by A10,A13,A17,JORDAN6:def 10;
end;
LE p1,p3,P by A1,A2,JORDAN6:73;
then LE p1,p4,P by A1,A2,JORDAN6:73;
then A18: p4=p1 by A2,A15,JORDAN6:72;
A19: LE p4,p2,P by A1,A2,A15,JORDAN6:73;
LE p2,p4,P by A1,A2,JORDAN6:73;
then A20: p2=p4 by A2,A19,JORDAN6:72;
LE p4,p3,P by A1,A2,A19,JORDAN6:73;
then p3=p4 by A1,A2,JORDAN6:72;
hence ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by A1,A11,A12,A18,A20,Th62;
case A21: p4`1<>-1;
then p4`1> -1 by A6,REAL_1:def 5;
then consider r being real number such that
A22: -1<r & r<p4`1 by REAL_1:75;
reconsider r1=r as Real by XREAL_0:def 1;
A23: -1<r1 & r1<1 by A6,A22,AXIOMS:22;
then consider f1 being map of TOP-REAL 2,TOP-REAL 2 such that
A24: f1=r1-FanMorphS & f1 is_homeomorphism by JGRAPH_4:143;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
now per cases;
case A25: (p4`1>0 or p4`2>=0);
then A26: p3`1>=0 or p3`2>=0 by A1,Th52;
then A27: p2`1>=0 or p2`2>=0 by A1,Th52;
then A28: p1`1>=0 or p1`2>=0 by A1,Th52;
now assume A29: p4`2=0 & p4`1<=0;
1^2 =(p4`1)^2+(p4`2)^2 by A5,JGRAPH_3:10
.=(p4`1)^2 by A29,SQUARE_1:60;
then p4`1=1 or p4`1=-1 by JGRAPH_3:1;
hence contradiction by A21,A29;
end;
hence ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by A1,A25,A26,A27,A28,Th66;
case A30: p4`1<=0 & p4`2<0;
p4`1/|.p4.|>r1 by A5,A22;
then A31: q44`1>0 & q44`2<0 by A23,A24,A30,Th29;
A32: LE q33,q44,P by A1,A23,A24,Th61;
A33: LE q22,q33,P by A1,A23,A24,Th61;
then A34: LE q22,q44,P by A2,A32,JORDAN6:73;
A35: LE q11,q22,P by A1,A23,A24,Th61;
then A36: LE q11,q44,P by A2,A34,JORDAN6:73;
W-min(P)=|[-1,0]| by A1,Th32;
then A37: (W-min(P))`2=0 by EUCLID:56;
A38: now per cases;
case q33`2>=0;
hence q33`2>=0 or q33`1>=0;
case q33`2<0;
then q33`1>=q44`1 by A1,A31,A32,A37,Th51;
hence q33`2>=0 or q33`1>=0 by A31,AXIOMS:22;
end;
A39: now per cases;
case q22`2>=0;
hence q22`2>=0 or q22`1>=0;
case q22`2<0;
then q22`1>=q44`1 by A1,A31,A34,A37,Th51;
hence q22`2>=0 or q22`1>=0 by A31,AXIOMS:22;
end;
now per cases;
case q11`2>=0;
hence q11`2>=0 or q11`1>=0;
case q11`2<0;
then q11`1>=q44`1 by A1,A31,A36,A37,Th51;
hence q11`2>=0 or q11`1>=0 by A31,AXIOMS:22;
end;
then consider f2 being map of TOP-REAL 2,TOP-REAL 2,
q81,q82,q83,q84 being Point of TOP-REAL 2 such that
A40: f2 is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.|)&
q81=f2.q11 & q82=f2.q22 & q83=f2.q33 & q84=f2.q44 &
(q81`1<0 & q81`2<0)&(q82`1<0 & q82`2<0)&(q83`1<0 & q83`2<0)
&(q84`1<0 & q84`2<0)&
LE q81,q82,P & LE q82,q83,P & LE q83,q84,P
by A1,A31,A32,A33,A35,A38,A39,Th66;
reconsider f3=f2*f1 as map of TOP-REAL 2,TOP-REAL 2;
A41: f3 is_homeomorphism by A24,A40,TOPS_2:71;
A42: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof let q be Point of TOP-REAL 2;
dom f1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then f3.q=f2.(f1.q) by FUNCT_1:23;
hence |.f3.q.|=|.(f1.q).| by A40 .=|.q.| by A24,JGRAPH_4:135;
end;
A43: dom f1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then A44: f3.p1=q81 by A40,FUNCT_1:23;
A45: f3.p2=q82 by A40,A43,FUNCT_1:23;
A46: f3.p3=q83 by A40,A43,FUNCT_1:23;
f3.p4=q84 by A40,A43,FUNCT_1:23;
hence ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P
by A40,A41,A42,A44,A45,A46;
end;
hence ex f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P;
end;
hence thesis;
end;
begin :: General Fashoda Meet Theorems
theorem Th69: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P &
p1<>p2 & p2<>p3 & p3<>p4 &
p1`1<0 & p2`1<0 & p3`1<0 & p4`1<0 &
p1`2<0 & p2`2<0 & p3`2<0 & p4`2<0
ex f being map of TOP-REAL 2,TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
|[-1,0]|=f.p1 & |[0,1]|=f.p2 & |[1,0]|=f.p3 & |[0,-1]|=f.p4
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P &
p1<>p2 & p2<>p3 & p3<>p4 &
p1`1<0 & p2`1<0 & p3`1<0 & p4`1<0 &
p1`2<0 & p2`2<0 & p3`2<0 & p4`2<0;
then A2: p1`1>p2`1 & p1`2<p2`2 by Th48;
A3: p2`1>p3`1 & p2`2<p3`2 by A1,Th48;
A4: p3`1>p4`1 & p3`2<p4`2 by A1,Th48;
P is_simple_closed_curve by A1,JGRAPH_3:36;
then A5: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& p1 in P & p2 in P & p3 in P & p4 in P &
LE p1,p2,P & LE p2,p3,P & LE p3,p4,P &
p1<>p2 & p2<>p3 & p3<>p4 &
p1`1<0 & p2`1<0 & p3`1<0 & p4`1<0 &
p1`2<0 & p2`2<0 & p3`2<0 & p4`2<0 by A1,JORDAN7:5;
then consider p11 being Point of TOP-REAL 2 such that
A6: p11=p1 & |.p11.|=1;
consider p22 being Point of TOP-REAL 2 such that
A7: p22=p2 & |.p22.|=1 by A5;
consider p33 being Point of TOP-REAL 2 such that
A8: p33=p3 & |.p33.|=1 by A5;
consider p44 being Point of TOP-REAL 2 such that
A9: p44=p4 & |.p44.|=1 by A5;
A10: -1<p1`2 & p1`2<1 by A1,A6,Th4;
then consider f1 being map of TOP-REAL 2,TOP-REAL 2 such that
A11: f1=((p1`2)-FanMorphW) & f1 is_homeomorphism by JGRAPH_4:48;
A12: p1`2/|.p1.|=p1`2 by A6;
A13: p2`2/|.p2.|=p2`2 by A7;
set q1=((p1`2)-FanMorphW).p1;
set q2=((p1`2)-FanMorphW).p2;
set q3=((p1`2)-FanMorphW).p3;
set q4=((p1`2)-FanMorphW).p4;
A14: |.q1.|=1 by A6,JGRAPH_4:40;
then A15: q1`2/|.q1.|=q1`2;
A16: |.q2.|=1 by A7,JGRAPH_4:40;
then A17: q2`2/|.q2.|=q2`2;
A18: |.q3.|=1 by A8,JGRAPH_4:40;
then A19: q3`2/|.q3.|=q3`2;
A20: |.q4.|=1 by A9,JGRAPH_4:40;
then A21: q4`2/|.q4.|=q4`2;
A22: q1`1<0 & q1`2=0 by A1,A10,A12,JGRAPH_4:54;
A23: q2`1<0 & q2`2>=0 by A1,A2,A10,A13,JGRAPH_4:49;
A24: p3`2/|.p3.|>p2`2 by A1,A8,Th48;
A25: p3`2/|.p3.|>p1`2 by A2,A3,A8,AXIOMS:22;
then A26: q3`1<0 & q3`2>=0 by A1,A10,JGRAPH_4:49;
A27: p4`2/|.p4.|>p3`2 by A1,A9,Th48;
p4`2/|.p4.|>p2`2 by A3,A4,A9,AXIOMS:22;
then A28: p4`2/|.p4.|>p1`2 by A2,AXIOMS:22;
A29: q1`2<q2`2 by A1,A2,A10,A12,A13,A15,A17,JGRAPH_4:51;
A30: q1`2<q2`2 & q2`2<q3`2 & q3`2<q4`2
by A1,A2,A8,A10,A12,A13,A15,A17,A19,A21,A24,A25,A27,A28,JGRAPH_4:51
;
A31: 0<q3`2 by A1,A10,A12,A15,A19,A22,A25,JGRAPH_4:51;
A32: 0<q4`2 by A1,A10,A12,A15,A21,A22,A28,JGRAPH_4:51;
A33: 1^2=(q2`1)^2+(q2`2)^2 by A16,JGRAPH_3:10;
A34: -q2`1>0 by A23,REAL_1:66;
A35: 1^2=(q3`1)^2+(q3`2)^2 by A18,JGRAPH_3:10;
then (q2`1)^2=(q3`1)^2+(q3`2)^2-(q2`2)^2 by A33,XCMPLX_1:26
.=(q3`1)^2+((q3`2)^2-(q2`2)^2) by XCMPLX_1:29;
then A36: (q2`1)^2-(q3`1)^2=(q3`2)^2-(q2`2)^2 by XCMPLX_1:26;
(q3`2)^2>(q2`2)^2 by A22,A30,SQUARE_1:78;
then (q3`2)^2-(q2`2)^2>0 by SQUARE_1:11;
then (q2`1)^2-(q3`1)^2+(q3`1)^2>0+(q3`1)^2 by A36,REAL_1:67;
then (q2`1)^2>(q3`1)^2 by XCMPLX_1:27;
then (-(q2`1))^2>(q3`1)^2 by SQUARE_1:61;
then A37: --(q2`1)<(q3`1) & q3`1<-(q2`1) by A34,JGRAPH_2:6;
A38: -q3`1>0 by A26,REAL_1:66;
1^2=(q4`1)^2+(q4`2)^2 by A20,JGRAPH_3:10;
then (q3`1)^2=(q4`1)^2+(q4`2)^2-(q3`2)^2 by A35,XCMPLX_1:26
.=(q4`1)^2+((q4`2)^2-(q3`2)^2) by XCMPLX_1:29;
then A39: (q3`1)^2-(q4`1)^2=(q4`2)^2-(q3`2)^2 by XCMPLX_1:26;
(q4`2)^2>(q3`2)^2 by A30,A31,SQUARE_1:78;
then (q4`2)^2-(q3`2)^2>0 by SQUARE_1:11;
then (q3`1)^2-(q4`1)^2+(q4`1)^2>0+(q4`1)^2 by A39,REAL_1:67;
then (q3`1)^2>(q4`1)^2 by XCMPLX_1:27;
then (-(q3`1))^2>(q4`1)^2 by SQUARE_1:61;
then A40: --(q3`1)<(q4`1) & q4`1<-(q3`1) by A38,JGRAPH_2:6;
(|.q1.|)^2 =(q1`1)^2+(q1`2)^2 by JGRAPH_3:10;
then A41: q1`1=-1 or q1`1=1 by A14,A22,JGRAPH_3:1,SQUARE_1:60;
then A42: q1=|[-1,0]| by A22,EUCLID:57;
A43: -1<q2`1 & q2`1<1 by A16,A22,A23,A30,Th4;
then consider f2 being map of TOP-REAL 2,TOP-REAL 2 such that
A44: f2=((q2`1)-FanMorphN) & f2 is_homeomorphism by JGRAPH_4:81;
A45: q2`1/|.q2.|=q2`1 by A16;
A46: q3`1/|.q3.|=q3`1 by A18;
set r1=((q2`1)-FanMorphN).q1;
set r2=((q2`1)-FanMorphN).q2;
set r3=((q2`1)-FanMorphN).q3;
set r4=((q2`1)-FanMorphN).q4;
A47: |.r2.|=1 by A16,JGRAPH_4:73;
then A48: r2`1/|.r2.|=r2`1;
A49: |.r3.|=1 by A18,JGRAPH_4:73;
then A50: r3`1/|.r3.|=r3`1;
A51: |.r4.|=1 by A20,JGRAPH_4:73;
then A52: r4`1/|.r4.|=r4`1;
A53: r2`2>0 & r2`1=0 by A22,A30,A43,A45,JGRAPH_4:87;
A54: r3`2>0 & r3`1>=0 by A31,A37,A43,A46,JGRAPH_4:82;
A55: q4`1/|.q4.|>q3`1 by A20,A40;
A56: q4`1/|.q4.|>q2`1 by A20,A37,A40,AXIOMS:22;
A57: r1=|[-1,0]| by A22,A42,JGRAPH_4:56;
A58: r1`1=-1 & r1`2=0 by A22,A41,JGRAPH_4:56;
(|.r2.|)^2 =(r2`1)^2+(r2`2)^2 by JGRAPH_3:10;
then A59: r2`2=-1 or r2`2=1 by A47,A53,JGRAPH_3:1,SQUARE_1:60;
then A60: r2=|[0,1]| by A53,EUCLID:57;
A61: r2`1<r3`1 by A22,A30,A31,A37,A43,A45,A46,A48,A50,JGRAPH_4:86;
A62: r2`1<r3`1 & r3`1<r4`1
by A22,A29,A31,A32,A37,A43,A45,A46,A48,A50,A52,A55,JGRAPH_4:86;
then A63: 0<r4`1 by A53,AXIOMS:22;
A64: r2`2>0 & r3`2>0 & r4`2>0 by A22,A29,A31,A32,A37,A43,A45,A46,A56,
JGRAPH_4:82;
A65: 1^2=(r2`1)^2+(r2`2)^2 by A47,JGRAPH_3:10;
A66: 1^2=(r3`1)^2+(r3`2)^2 by A49,JGRAPH_3:10;
then (r2`2)^2=(r3`2)^2+(r3`1)^2-(r2`1)^2 by A65,XCMPLX_1:26
.=(r3`2)^2+((r3`1)^2-(r2`1)^2) by XCMPLX_1:29;
then A67: (r2`2)^2-(r3`2)^2=(r3`1)^2-(r2`1)^2 by XCMPLX_1:26;
(r3`1)^2>(r2`1)^2 by A53,A62,SQUARE_1:78;
then (r3`1)^2-(r2`1)^2>0 by SQUARE_1:11;
then (r2`2)^2-(r3`2)^2+(r3`2)^2>0+(r3`2)^2 by A67,REAL_1:67;
then A68: (r2`2)^2>(r3`2)^2 by XCMPLX_1:27;
1^2=(r4`1)^2+(r4`2)^2 by A51,JGRAPH_3:10;
then (r3`2)^2=(r4`2)^2+(r4`1)^2-(r3`1)^2 by A66,XCMPLX_1:26
.=(r4`2)^2+((r4`1)^2-(r3`1)^2) by XCMPLX_1:29;
then A69: (r3`2)^2-(r4`2)^2=(r4`1)^2-(r3`1)^2 by XCMPLX_1:26;
(r4`1)^2>(r3`1)^2 by A53,A62,SQUARE_1:78;
then (r4`1)^2-(r3`1)^2>0 by SQUARE_1:11;
then (r3`2)^2-(r4`2)^2+(r4`2)^2>0+(r4`2)^2 by A69,REAL_1:67;
then ((r3`2))^2>(r4`2)^2 by XCMPLX_1:27;
then A70: r2`2>r3`2 & r3`2>r4`2 by A64,A68,JGRAPH_2:6;
A71: -1<r3`2 & r3`2<1 by A49,A53,A54,A61,Th4;
then consider f3 being map of TOP-REAL 2,TOP-REAL 2 such that
A72: f3=((r3`2)-FanMorphE) & f3 is_homeomorphism by JGRAPH_4:112;
A73: r3`2/|.r3.|=r3`2 by A49;
A74: r4`2/|.r4.|=r4`2 by A51;
set s1=((r3`2)-FanMorphE).r1;
set s2=((r3`2)-FanMorphE).r2;
set s3=((r3`2)-FanMorphE).r3;
set s4=((r3`2)-FanMorphE).r4;
A75: |.s3.|=1 by A49,JGRAPH_4:104;
A76: |.s4.|=1 by A51,JGRAPH_4:104;
A77: s3`1>0 & s3`2=0 by A53,A62,A71,A73,JGRAPH_4:118;
A78: s4`1>0 & s4`2<0 by A63,A70,A71,A74,JGRAPH_4:114;
A79: s1=|[-1,0]| by A57,A58,JGRAPH_4:89;
A80: s1`1=-1 & s1`2=0 by A58,JGRAPH_4:89;
A81: s2=|[0,1]| by A53,A60,JGRAPH_4:89;
A82: s2`1=0 & s2`2=1 by A53,A59,JGRAPH_4:89;
(|.s3.|)^2 =(s3`1)^2+(s3`2)^2 by JGRAPH_3:10;
then s3`1=-1 or s3`1=1 by A75,A77,JGRAPH_3:1,SQUARE_1:60;
then A83: s3=|[1,0]| by A77,EUCLID:57;
A84: s3`2/|.s3.|>s4`2/|.s4.| by A53,A62,A63,A70,A71,A73,A74,JGRAPH_4:117;
A85: -1<s4`1 & s4`1<1 by A76,A78,Th4;
then consider f4 being map of TOP-REAL 2,TOP-REAL 2 such that
A86: f4=((s4`1)-FanMorphS) & f4 is_homeomorphism by JGRAPH_4:143;
A87: s4`1/|.s4.|=s4`1 by A76;
set t4=((s4`1)-FanMorphS).s4;
A88: |.t4.|=1 by A76,JGRAPH_4:135;
A89: t4`2<0 & t4`1=0 by A76,A77,A84,A85,A87,JGRAPH_4:149;
A90: ((s4`1)-FanMorphS).s1=|[-1,0]| by A79,A80,JGRAPH_4:120;
A91: ((s4`1)-FanMorphS).s2=|[0,1]| by A81,A82,JGRAPH_4:120;
A92: ((s4`1)-FanMorphS).s3=|[1,0]| by A77,A83,JGRAPH_4:120;
(|.t4.|)^2 =(t4`1)^2+(t4`2)^2 by JGRAPH_3:10;
then t4`2=-1 or t4`2=1 by A88,A89,JGRAPH_3:1,SQUARE_1:60;
then A93: t4=|[0,-1]| by A89,EUCLID:57;
reconsider g=f4*(f3*(f2*f1)) as map of TOP-REAL 2,TOP-REAL 2;
f2*f1 is_homeomorphism by A11,A44,TOPS_2:71;
then f3*(f2*f1) is_homeomorphism by A72,TOPS_2:71;
then A94: g is_homeomorphism by A86,TOPS_2:71;
A95: dom g=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A96: dom (f2*f1)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A97: dom (f3*(f2*f1))=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A98: for q being Point of TOP-REAL 2 holds |.(g.q).|=|.q.|
proof let q be Point of TOP-REAL 2;
A99: |.((f2*f1).q).|=|.(f2.(f1.q)).| by A96,FUNCT_1:22
.=|.(f1.q).| by A44,JGRAPH_4:73 .=|.q.| by A11,JGRAPH_4:40;
A100: |.((f3*(f2*f1)).q).|=|.(f3.((f2*f1).q)).| by A97,FUNCT_1:22
.=|.q.| by A72,A99,JGRAPH_4:104;
thus |.(g.q).|=|.(f4.((f3*(f2*f1)).q)).| by A95,FUNCT_1:22
.=|.q.| by A86,A100,JGRAPH_4:135;
end;
A101: g.p1=(f4.((f3*(f2*f1)).p1)) by A95,FUNCT_1:22
.=f4.((f3.((f2*f1).p1))) by A97,FUNCT_1:22
.=|[-1,0]| by A11,A44,A72,A86,A90,A96,FUNCT_1:22;
A102: g.p2=(f4.((f3*(f2*f1)).p2)) by A95,FUNCT_1:22
.=f4.((f3.((f2*f1).p2))) by A97,FUNCT_1:22
.=|[0,1]| by A11,A44,A72,A86,A91,A96,FUNCT_1:22;
A103: g.p3= (f4.((f3*(f2*f1)).p3)) by A95,FUNCT_1:22
.=f4.((f3.((f2*f1).p3))) by A97,FUNCT_1:22
.=|[1,0]| by A11,A44,A72,A86,A92,A96,FUNCT_1:22;
g.p4= (f4.((f3*(f2*f1)).p4)) by A95,FUNCT_1:22
.=f4.((f3.((f2*f1).p4))) by A97,FUNCT_1:22
.=|[0,-1]| by A11,A44,A72,A86,A93,A96,FUNCT_1:22;
hence thesis by A94,A98,A101,A102,A103;
end;
theorem Th70: for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P &
p1<>p2 & p2<>p3 & p3<>p4
ex f being map of TOP-REAL 2,TOP-REAL 2 st
f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
|[-1,0]|=f.p1 & |[0,1]|=f.p2 & |[1,0]|=f.p3 & |[0,-1]|=f.p4
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P &
p1<>p2 & p2<>p3 & p3<>p4;
then consider f being map of TOP-REAL 2,TOP-REAL 2,
q1,q2,q3,q4 being Point of TOP-REAL 2 such that
A2: f is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)&
q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 &
(q1`1<0 & q1`2<0)&(q2`1<0 & q2`2<0)&
(q3`1<0 & q3`2<0)&(q4`1<0 & q4`2<0)&
LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by Th68;
A3: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A4: f is one-to-one by A2,TOPS_2:def 5;
then A5: q1<>q2 by A1,A2,A3,FUNCT_1:def 8;
A6: q2<>q3 by A1,A2,A3,A4,FUNCT_1:def 8;
q3<>q4 by A1,A2,A3,A4,FUNCT_1:def 8;
then consider f2 being map of TOP-REAL 2,TOP-REAL 2 such that
A7: f2 is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.|)&
|[-1,0]|=f2.q1 & |[0,1]|=f2.q2 & |[1,0]|=f2.q3 & |[0,-1]|=f2.q4
by A1,A2,A5,A6,Th69;
reconsider f3=f2*f as map of TOP-REAL 2,TOP-REAL 2;
A8: f3 is_homeomorphism by A2,A7,TOPS_2:71;
A9: dom f3=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A10: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof let q be Point of TOP-REAL 2;
|.(f3.q).|=|.f2.(f.q).| by A9,FUNCT_1:22.=|.(f.q).| by A7
.=|.q.| by A2;
hence |.(f3.q).|=|.q.|;
end;
A11: f3.p1=|[-1,0]| by A2,A7,A9,FUNCT_1:22;
A12: f3.p2=|[0,1]| by A2,A7,A9,FUNCT_1:22;
A13: f3.p3=|[1,0]| by A2,A7,A9,FUNCT_1:22;
f3.p4=|[0,-1]| by A2,A7,A9,FUNCT_1:22;
hence thesis by A8,A10,A11,A12,A13;
end;
Lm6: (|[-1,0]|)`1 =-1 & (|[-1,0]|)`2=0 & (|[1,0]|)`1 =1 & (|[1,0]|)`2=0 &
(|[0,-1]|)`1 =0 & (|[0,-1]|)`2=-1 & (|[0,1]|)`1 =0 & (|[0,1]|)`2=1
by EUCLID:56;
Lm7: now thus |.(|[-1,0]|).|=sqrt((-1)^2+0^2) by Lm6,JGRAPH_3:10
.=1 by SQUARE_1:59,60,61,83;
thus |.(|[1,0]|).|=sqrt(1+0) by Lm6,JGRAPH_3:10,SQUARE_1:59,60
.=1 by SQUARE_1:83;
thus |.(|[0,-1]|).|=sqrt(0^2+(-1)^2) by Lm6,JGRAPH_3:10
.=1 by SQUARE_1:59,60,61,83;
thus |.(|[0,1]|).|=sqrt(0^2+1^2) by Lm6,JGRAPH_3:10
.=1 by SQUARE_1:59,60,83;
end;
Lm8: 0 in [.0,1.] by TOPREAL5:1;
Lm9: 1 in [.0,1.] by TOPREAL5:1;
theorem for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
holds
(for f,g being map of I[01],TOP-REAL 2 st
f is continuous one-to-one &
g is continuous one-to-one &
C0={p: |.p.|<=1}& f.0=p1 & f.1=p3 & g.0=p2 & g.1=p4 &
rng f c= C0 & rng g c= C0 holds rng f meets rng g)
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P;
let f,g be map of I[01],TOP-REAL 2;
assume A2: f is continuous one-to-one &
g is continuous one-to-one &
C0={p: |.p.|<=1}& f.0=p1 & f.1=p3 & g.0=p2 & g.1=p4 &
rng f c= C0 & rng g c= C0;
A3: dom f=the carrier of I[01] by FUNCT_2:def 1;
A4: dom g=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now per cases by A5;
case A6: p1=p2;
thus rng f meets rng g
proof
A7: p1 in rng f by A2,A3,Lm8,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm8,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A6,A7,XBOOLE_0:3;
end;
case A8: p2=p3;
thus rng f meets rng g
proof
A9: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm8,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A8,A9,XBOOLE_0:3;
end;
case A10: p3=p4;
thus rng f meets rng g
proof
A11: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p4 in rng g by A2,A4,Lm9,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A10,A11,XBOOLE_0:3;
end;
end;
hence thesis;
suppose p1<>p2 & p2<>p3 & p3<>p4;
then consider h being map of TOP-REAL 2,TOP-REAL 2 such that
A12: h is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.|)&
|[-1,0]|=h.p1 & |[0,1]|=h.p2 & |[1,0]|=h.p3 & |[0,-1]|=h.p4
by A1,Th70;
A13: h is one-to-one by A12,TOPS_2:def 5;
reconsider f2=h*f as map of I[01],TOP-REAL 2;
reconsider g2=h*g as map of I[01],TOP-REAL 2;
A14: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A15: dom g2=the carrier of I[01] by FUNCT_2:def 1;
A16: f2.0= |[-1,0]| by A2,A12,A14,Lm8,BORSUK_1:83,FUNCT_1:22;
A17: g2.0= |[0,1]| by A2,A12,A15,Lm8,BORSUK_1:83,FUNCT_1:22;
A18: f2.1= |[1,0]| by A2,A12,A14,Lm9,BORSUK_1:83,FUNCT_1:22;
A19: g2.1= |[0,-1]| by A2,A12,A15,Lm9,BORSUK_1:83,FUNCT_1:22;
A20: f2 is continuous one-to-one &
g2 is continuous one-to-one &
f2.0=|[-1,0]| & f2.1=|[1,0]| & g2.0=|[0,1]| & g2.1= |[0,-1]|
by A2,A12,A14,A15,Lm8,Lm9,Th8,Th9,BORSUK_1:83,
FUNCT_1:22;
A21: rng f2 c= C0
proof let y be set;assume y in rng f2;
then consider x being set such that
A22: x in dom f2 & y=f2.x by FUNCT_1:def 5;
A23: f2.x=h.(f.x) by A22,FUNCT_1:22;
A24: f.x in rng f by A3,A14,A22,FUNCT_1:def 5;
then A25: f.x in C0 by A2;
reconsider qf=f.x as Point of TOP-REAL 2 by A24;
consider q5 being Point of TOP-REAL 2 such that
A26: q5=f.x & |.q5.|<=1 by A2,A25;
|.(h.qf).|=|.qf.| by A12;
hence y in C0 by A2,A22,A23,A26;
end;
A27: rng g2 c= C0
proof let y be set;assume y in rng g2;
then consider x being set such that
A28: x in dom g2 & y=g2.x by FUNCT_1:def 5;
A29: g2.x=h.(g.x) by A28,FUNCT_1:22;
A30: g.x in rng g by A4,A15,A28,FUNCT_1:def 5;
then A31: g.x in C0 by A2;
reconsider qg=g.x as Point of TOP-REAL 2 by A30;
consider q5 being Point of TOP-REAL 2 such that
A32: q5=g.x & |.q5.|<=1 by A2,A31;
|.(h.qg).|=|.qg.| by A12;
hence y in C0 by A2,A28,A29,A32;
end;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} as Subset of TOP-REAL 2;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:83,TOPREAL5:1;
-(|[-1,0]|)`1=1 by Lm6;
then A33: f2.O in KXN by A16,Lm6,Lm7;
A34: f2.I in KXP by A18,Lm6,Lm7;
-(|[0,-1]|)`1= 0 by Lm6;
then A35: g2.I in KYN by A19,Lm6,Lm7;
-(|[0,1]|)`1= 0 by Lm6;
then g2.O in KYP by A17,Lm6,Lm7;
then rng f2 meets rng g2 by A2,A20,A21,A27,A33,A34,A35,Th16;
then consider x2 being set such that
A36: x2 in rng f2 & x2 in rng g2 by XBOOLE_0:3;
consider z2 being set such that
A37: z2 in dom f2 & x2=f2.z2 by A36,FUNCT_1:def 5;
consider z3 being set such that
A38: z3 in dom g2 & x2=g2.z3 by A36,FUNCT_1:def 5;
A39: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A40: g.z3 in rng g by A4,A15,A38,FUNCT_1:def 5;
A41: f.z2 in rng f by A3,A14,A37,FUNCT_1:def 5;
reconsider h1=h as Function;
A42: h1".x2=h1".(h.(f.z2)) by A37,FUNCT_1:22 .=f.z2
by A13,A39,A41,FUNCT_1:56;
h1".x2=h1".(h.(g.z3)) by A38,FUNCT_1:22 .=g.z3
by A13,A39,A40,FUNCT_1:56;
then h1".x2 in rng f & h1".x2 in rng g by A3,A4,A14,A15,A37,A38,A42,FUNCT_1
:def 5;
hence thesis by XBOOLE_0:3;
end;
theorem
for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P
holds
(for f,g being map of I[01],TOP-REAL 2 st
f is continuous one-to-one & g is continuous one-to-one &
C0={p: |.p.|<=1}& f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 &
rng f c= C0 & rng g c= C0 holds rng f meets rng g)
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P;
let f,g be map of I[01],TOP-REAL 2;
assume A2:f is continuous one-to-one & g is continuous one-to-one &
C0={p: |.p.|<=1}& f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 &
rng f c= C0 & rng g c= C0;
A3: dom f=the carrier of I[01] by FUNCT_2:def 1;
A4: dom g=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now per cases by A5;
case A6: p1=p2;
thus rng f meets rng g
proof
A7: p1 in rng f by A2,A3,Lm8,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm9,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A6,A7,XBOOLE_0:3;
end;
case A8: p2=p3;
thus rng f meets rng g
proof
A9: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm9,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A8,A9,XBOOLE_0:3;
end;
case A10: p3=p4;
thus rng f meets rng g
proof
A11: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p4 in rng g by A2,A4,Lm8,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A10,A11,XBOOLE_0:3;
end;
end;
hence thesis;
suppose p1<>p2 & p2<>p3 & p3<>p4;
then consider h being map of TOP-REAL 2,TOP-REAL 2 such that
A12: h is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.|)&
|[-1,0]|=h.p1 & |[0,1]|=h.p2 & |[1,0]|=h.p3 & |[0,-1]|=h.p4
by A1,Th70;
A13: h is one-to-one by A12,TOPS_2:def 5;
reconsider f2=h*f as map of I[01],TOP-REAL 2;
reconsider g2=h*g as map of I[01],TOP-REAL 2;
A14: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A15: dom g2=the carrier of I[01] by FUNCT_2:def 1;
A16: f2.0= |[-1,0]| by A2,A12,A14,Lm8,BORSUK_1:83,FUNCT_1:22;
A17: g2.0= |[0,-1]| by A2,A12,A15,Lm8,BORSUK_1:83,FUNCT_1:22;
A18: f2.1= |[1,0]| by A2,A12,A14,Lm9,BORSUK_1:83,FUNCT_1:22;
A19: g2.1= |[0,1]| by A2,A12,A15,Lm9,BORSUK_1:83,FUNCT_1:22;
A20: f2 is continuous one-to-one &
g2 is continuous one-to-one &
f2.0=|[-1,0]| & f2.1=|[1,0]| & g2.0=|[0,-1]| & g2.1= |[0,1]|
by A2,A12,A14,A15,Lm8,Lm9,Th8,Th9,BORSUK_1:83,
FUNCT_1:22;
A21: rng f2 c= C0
proof let y be set;assume y in rng f2;
then consider x being set such that
A22: x in dom f2 & y=f2.x by FUNCT_1:def 5;
A23: f2.x=h.(f.x) by A22,FUNCT_1:22;
A24: f.x in rng f by A3,A14,A22,FUNCT_1:def 5;
then A25: f.x in C0 by A2;
reconsider qf=f.x as Point of TOP-REAL 2 by A24;
consider q5 being Point of TOP-REAL 2 such that
A26: q5=f.x & |.q5.|<=1 by A2,A25;
|.(h.qf).|=|.qf.| by A12;
hence y in C0 by A2,A22,A23,A26;
end;
A27: rng g2 c= C0
proof let y be set;assume y in rng g2;
then consider x being set such that
A28: x in dom g2 & y=g2.x by FUNCT_1:def 5;
A29: g2.x=h.(g.x) by A28,FUNCT_1:22;
A30: g.x in rng g by A4,A15,A28,FUNCT_1:def 5;
then A31: g.x in C0 by A2;
reconsider qg=g.x as Point of TOP-REAL 2 by A30;
consider q5 being Point of TOP-REAL 2 such that
A32: q5=g.x & |.q5.|<=1 by A2,A31;
|.(h.qg).|=|.qg.| by A12;
hence y in C0 by A2,A28,A29,A32;
end;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} as Subset of TOP-REAL 2;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:83,TOPREAL5:1;
-(|[-1,0]|)`1=1 by Lm6;
then A33: f2.O in KXN by A16,Lm6,Lm7;
A34: f2.I in KXP by A18,Lm6,Lm7;
-(|[0,-1]|)`1= 0 by Lm6;
then A35: g2.O in KYN by A17,Lm6,Lm7;
-(|[0,1]|)`1= 0 by Lm6;
then g2.I in KYP by A19,Lm6,Lm7;
then rng f2 meets rng g2 by A2,A20,A21,A27,A33,A34,A35,JGRAPH_3:55;
then consider x2 being set such that
A36: x2 in rng f2 & x2 in rng g2 by XBOOLE_0:3;
consider z2 being set such that
A37: z2 in dom f2 & x2=f2.z2 by A36,FUNCT_1:def 5;
consider z3 being set such that
A38: z3 in dom g2 & x2=g2.z3 by A36,FUNCT_1:def 5;
A39: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A40: g.z3 in rng g by A4,A15,A38,FUNCT_1:def 5;
A41: f.z2 in rng f by A3,A14,A37,FUNCT_1:def 5;
reconsider h1=h as Function;
A42: h1".x2=h1".(h.(f.z2)) by A37,FUNCT_1:22 .=f.z2
by A13,A39,A41,FUNCT_1:56;
h1".x2=h1".(h.(g.z3)) by A38,FUNCT_1:22 .=g.z3
by A13,A39,A40,FUNCT_1:56;
then h1".x2 in rng f & h1".x2 in rng g by A3,A4,A14,A15,A37,A38,A42,FUNCT_1
:def 5;
hence thesis by XBOOLE_0:3;
end;
theorem for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
(for f,g being map of I[01],TOP-REAL 2 st
f is continuous one-to-one & g is continuous one-to-one &
C0={p: |.p.|>=1}& f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 &
rng f c= C0 & rng g c= C0 holds rng f meets rng g)
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P;
let f,g be map of I[01],TOP-REAL 2;
assume A2: f is continuous one-to-one & g is continuous one-to-one &
C0={p: |.p.|>=1}& f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 &
rng f c= C0 & rng g c= C0;
A3: dom f=the carrier of I[01] by FUNCT_2:def 1;
A4: dom g=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now per cases by A5;
case A6: p1=p2;
thus rng f meets rng g
proof
A7: p1 in rng f by A2,A3,Lm8,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm9,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A6,A7,XBOOLE_0:3;
end;
case A8: p2=p3;
thus rng f meets rng g
proof
A9: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm9,BORSUK_1:83,FUNCT_1:def 5;
hence rng f meets rng g by A8,A9,XBOOLE_0:3;
end;
case A10: p3=p4;
thus rng f meets rng g
proof
A11: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p4 in rng g by A2,A4,Lm8,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A10,A11,XBOOLE_0:3;
end;
end;
hence thesis;
suppose p1<>p2 & p2<>p3 & p3<>p4;
then consider h being map of TOP-REAL 2,TOP-REAL 2 such that
A12: h is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.|)&
|[-1,0]|=h.p1 & |[0,1]|=h.p2 & |[1,0]|=h.p3 & |[0,-1]|=h.p4
by A1,Th70;
A13: h is one-to-one by A12,TOPS_2:def 5;
reconsider f2=h*f as map of I[01],TOP-REAL 2;
reconsider g2=h*g as map of I[01],TOP-REAL 2;
A14: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A15: dom g2=the carrier of I[01] by FUNCT_2:def 1;
A16: f2.0= |[-1,0]| by A2,A12,A14,Lm8,BORSUK_1:83,FUNCT_1:22;
A17: g2.0= |[0,-1]| by A2,A12,A15,Lm8,BORSUK_1:83,FUNCT_1:22;
A18: f2.1= |[1,0]| by A2,A12,A14,Lm9,BORSUK_1:83,FUNCT_1:22;
A19: g2.1= |[0,1]| by A2,A12,A15,Lm9,BORSUK_1:83,FUNCT_1:22;
A20: f2 is continuous one-to-one & g2 is continuous one-to-one &
f2.0=|[-1,0]| & f2.1=|[1,0]| & g2.0=|[0,-1]| & g2.1= |[0,1]|
by A2,A12,A14,A15,Lm8,Lm9,Th8,Th9,BORSUK_1:83,
FUNCT_1:22;
A21: rng f2 c= C0
proof let y be set;assume y in rng f2;
then consider x being set such that
A22: x in dom f2 & y=f2.x by FUNCT_1:def 5;
A23: f2.x=h.(f.x) by A22,FUNCT_1:22;
A24: f.x in rng f by A3,A14,A22,FUNCT_1:def 5;
then A25: f.x in C0 by A2;
reconsider qf=f.x as Point of TOP-REAL 2 by A24;
consider q5 being Point of TOP-REAL 2 such that
A26: q5=f.x & |.q5.|>=1 by A2,A25;
|.(h.qf).|=|.qf.| by A12;
hence y in C0 by A2,A22,A23,A26;
end;
A27: rng g2 c= C0
proof let y be set;assume y in rng g2;
then consider x being set such that
A28: x in dom g2 & y=g2.x by FUNCT_1:def 5;
A29: g2.x=h.(g.x) by A28,FUNCT_1:22;
A30: g.x in rng g by A4,A15,A28,FUNCT_1:def 5;
then A31: g.x in C0 by A2;
reconsider qg=g.x as Point of TOP-REAL 2 by A30;
consider q5 being Point of TOP-REAL 2 such that
A32: q5=g.x & |.q5.|>=1 by A2,A31;
|.(h.qg).|=|.qg.| by A12;
hence y in C0 by A2,A28,A29,A32;
end;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} as Subset of TOP-REAL 2;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:83,TOPREAL5:1;
-(|[-1,0]|)`1=1 by Lm6;
then A33: f2.O in KXN by A16,Lm6,Lm7;
A34: f2.I in KXP by A18,Lm6,Lm7;
-(|[0,-1]|)`1= 0 by Lm6;
then A35: g2.O in KYN by A17,Lm6,Lm7;
-(|[0,1]|)`1= 0 by Lm6;
then g2.I in KYP by A19,Lm6,Lm7;
then rng f2 meets rng g2 by A2,A20,A21,A27,A33,A34,A35,Th17;
then consider x2 being set such that
A36: x2 in rng f2 & x2 in rng g2 by XBOOLE_0:3;
consider z2 being set such that
A37: z2 in dom f2 & x2=f2.z2 by A36,FUNCT_1:def 5;
consider z3 being set such that
A38: z3 in dom g2 & x2=g2.z3 by A36,FUNCT_1:def 5;
A39: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A40: g.z3 in rng g by A4,A15,A38,FUNCT_1:def 5;
A41: f.z2 in rng f by A3,A14,A37,FUNCT_1:def 5;
reconsider h1=h as Function;
A42: h1".x2=h1".(h.(f.z2)) by A37,FUNCT_1:22 .=f.z2
by A13,A39,A41,FUNCT_1:56;
h1".x2=h1".(h.(g.z3)) by A38,FUNCT_1:22 .=g.z3
by A13,A39,A40,FUNCT_1:56;
then h1".x2 in rng f & h1".x2 in rng g by A3,A4,A14,A15,A37,A38,A42,FUNCT_1
:def 5;
hence rng f meets rng g by XBOOLE_0:3;
end;
theorem for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
(for f,g being map of I[01],TOP-REAL 2 st
f is continuous one-to-one & g is continuous one-to-one &
C0={p: |.p.|>=1}& f.0=p1 & f.1=p3 & g.0=p2 & g.1=p4 &
rng f c= C0 & rng g c= C0 holds rng f meets rng g)
proof let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be compact non empty Subset of TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume A1: P={p where p is Point of TOP-REAL 2: |.p.|=1}
& LE p1,p2,P & LE p2,p3,P & LE p3,p4,P;
let f,g be map of I[01],TOP-REAL 2;
assume A2: f is continuous one-to-one & g is continuous one-to-one &
C0={p: |.p.|>=1}& f.0=p1 & f.1=p3 & g.0=p2 & g.1=p4 &
rng f c= C0 & rng g c= C0;
A3: dom f=the carrier of I[01] by FUNCT_2:def 1;
A4: dom g=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now per cases by A5;
case A6: p1=p2;
thus rng f meets rng g
proof
A7: p1 in rng f by A2,A3,Lm8,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm8,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A6,A7,XBOOLE_0:3;
end;
case A8: p2=p3;
thus rng f meets rng g
proof
A9: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p2 in rng g by A2,A4,Lm8,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A8,A9,XBOOLE_0:3;
end;
case A10: p3=p4;
thus rng f meets rng g
proof
A11: p3 in rng f by A2,A3,Lm9,BORSUK_1:83,FUNCT_1:def 5;
p4 in rng g by A2,A4,Lm9,BORSUK_1:83,FUNCT_1:def 5;
hence thesis by A10,A11,XBOOLE_0:3;
end;
end;
hence thesis;
suppose p1<>p2 & p2<>p3 & p3<>p4;
then consider h being map of TOP-REAL 2,TOP-REAL 2 such that
A12: h is_homeomorphism &
(for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.|)&
|[-1,0]|=h.p1 & |[0,1]|=h.p2 & |[1,0]|=h.p3 & |[0,-1]|=h.p4
by A1,Th70;
A13: h is one-to-one by A12,TOPS_2:def 5;
reconsider f2=h*f,g2=h*g as map of I[01],TOP-REAL 2;
A14: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A15: dom g2=the carrier of I[01] by FUNCT_2:def 1;
A16: f2.0 = |[-1,0]| by A2,A12,A14,Lm8,BORSUK_1:83,FUNCT_1:22;
A17: g2.0 = |[0,1]| by A2,A12,A15,Lm8,BORSUK_1:83,FUNCT_1:22;
A18: f2.1 = |[1,0]| by A2,A12,A14,Lm9,BORSUK_1:83,FUNCT_1:22;
A19: g2.1 = |[0,-1]| by A2,A12,A15,Lm9,BORSUK_1:83,FUNCT_1:22;
A20: f2 is continuous one-to-one & g2 is continuous one-to-one &
f2.0=|[-1,0]| & f2.1=|[1,0]| & g2.0=|[0,1]| & g2.1= |[0,-1]|
by A2,A12,A14,A15,Lm8,Lm9,Th8,Th9,BORSUK_1:83,
FUNCT_1:22;
A21: rng f2 c= C0
proof let y be set;assume y in rng f2;
then consider x being set such that
A22: x in dom f2 & y=f2.x by FUNCT_1:def 5;
A23: f2.x=h.(f.x) by A22,FUNCT_1:22;
A24: f.x in rng f by A3,A14,A22,FUNCT_1:def 5;
then A25: f.x in C0 by A2;
reconsider qf=f.x as Point of TOP-REAL 2 by A24;
consider q5 being Point of TOP-REAL 2 such that
A26: q5=f.x & |.q5.|>=1 by A2,A25;
|.(h.qf).|=|.qf.| by A12;
hence y in C0 by A2,A22,A23,A26;
end;
A27: rng g2 c= C0
proof let y be set;assume y in rng g2;
then consider x being set such that
A28: x in dom g2 & y=g2.x by FUNCT_1:def 5;
A29: g2.x=h.(g.x) by A28,FUNCT_1:22;
A30: g.x in rng g by A4,A15,A28,FUNCT_1:def 5;
then A31: g.x in C0 by A2;
reconsider qg=g.x as Point of TOP-REAL 2 by A30;
consider q5 being Point of TOP-REAL 2 such that
A32: q5=g.x & |.q5.|>=1 by A2,A31;
|.(h.qg).|=|.qg.| by A12;
hence y in C0 by A2,A28,A29,A32;
end;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXP={q1 where q1 is Point of TOP-REAL 2:
|.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KXN={q2 where q2 is Point of TOP-REAL 2:
|.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYP={q3 where q3 is Point of TOP-REAL 2:
|.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means
|.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is
Subset of TOP-REAL 2 from TopSubset;
then reconsider KYN={q4 where q4 is Point of TOP-REAL 2:
|.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} as Subset of TOP-REAL 2;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:83,TOPREAL5:1;
-(|[-1,0]|)`1=1 by Lm6;
then A33: f2.O in KXN by A16,Lm6,Lm7;
A34: f2.I in KXP by A18,Lm6,Lm7;
-(|[0,-1]|)`1= 0 by Lm6;
then A35: g2.I in KYN by A19,Lm6,Lm7;
-(|[0,1]|)`1= 0 by Lm6;
then g2.O in KYP by A17,Lm6,Lm7;
then rng f2 meets rng g2 by A2,A20,A21,A27,A33,A34,A35,Th18;
then consider x2 being set such that
A36: x2 in rng f2 & x2 in rng g2 by XBOOLE_0:3;
consider z2 being set such that
A37: z2 in dom f2 & x2=f2.z2 by A36,FUNCT_1:def 5;
consider z3 being set such that
A38: z3 in dom g2 & x2=g2.z3 by A36,FUNCT_1:def 5;
A39: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A40: g.z3 in rng g by A4,A15,A38,FUNCT_1:def 5;
A41: f.z2 in rng f by A3,A14,A37,FUNCT_1:def 5;
reconsider h1=h as Function;
A42: h1".x2=h1".(h.(f.z2)) by A37,FUNCT_1:22 .=f.z2 by A13,A39,A41,FUNCT_1:
56;
h1".x2=h1".(h.(g.z3)) by A38,FUNCT_1:22 .=g.z3
by A13,A39,A40,FUNCT_1:56;
then h1".x2 in rng f & h1".x2 in rng g by A3,A4,A14,A15,A37,A38,A42,FUNCT_1
:def 5;
hence rng f meets rng g by XBOOLE_0:3;
end;