### The Mizar article:

### On Outside Fashoda Meet Theorem

**by****Yatsuka Nakamura**

- Received July 16, 2001
Copyright (c) 2001 Association of Mizar Users

- MML identifier: JGRAPH_2
- [ MML identifier index ]

environ vocabulary EUCLID, PCOMPS_1, ARYTM, ARYTM_3, RELAT_1, SQUARE_1, ARYTM_1, PRE_TOPC, SUBSET_1, BOOLE, ORDINAL2, FUNCT_1, FUNCT_4, METRIC_1, COMPLEX1, MCART_1, JORDAN2C, FINSEQ_1, FINSEQ_2, FUNCT_5, TOPMETR, RCOMP_1, PARTFUN1, BORSUK_1, JGRAPH_2; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, RELAT_1, FUNCT_1, FINSEQ_1, FINSEQ_2, NAT_1, STRUCT_0, PARTFUN1, PRE_TOPC, TOPMETR, PCOMPS_1, METRIC_1, RCOMP_1, FUNCT_2, SQUARE_1, PSCOMP_1, EUCLID, JGRAPH_1, JORDAN2C, FUNCT_4, WELLFND1; constructors REAL_1, WEIERSTR, TOPS_2, RCOMP_1, PSCOMP_1, JORDAN2C, WELLFND1, FUNCT_4, TOPRNS_1, MEMBERED; clusters SUBSET_1, STRUCT_0, RELSET_1, EUCLID, PRE_TOPC, TOPMETR, SQUARE_1, PSCOMP_1, BORSUK_1, XREAL_0, ARYTM_3, MEMBERED, ZFMISC_1; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions TARSKI, XBOOLE_0; theorems TARSKI, AXIOMS, RELAT_1, SUBSET_1, FUNCT_1, FUNCT_2, FUNCT_4, TOPS_1, TOPS_2, PARTFUN1, PRE_TOPC, REVROT_1, JORDAN2C, FINSEQ_2, FRECHET, TOPMETR, JORDAN6, EUCLID, REAL_1, REAL_2, JGRAPH_1, SEQ_2, SQUARE_1, TOPREAL3, TOPREAL6, PSCOMP_1, METRIC_1, SPPOL_2, JORDAN1A, TSEP_1, XBOOLE_0, XBOOLE_1, XREAL_0, XCMPLX_0, XCMPLX_1; schemes FUNCT_1, FUNCT_2; begin Lm1:TOP-REAL 2=TopSpaceMetr(Euclid 2) by EUCLID:def 8; canceled; theorem Th2: for a being real number st 1 <= a holds a <= a^2 proof let a be real number;assume A1: 1 <= a; then a>=0 by AXIOMS:22; then a <= a*a by A1,REAL_2:146; hence a <= a^2 by SQUARE_1:def 3; end; theorem for a being real number st -1 >= a holds -a <= a^2 proof let a be real number;assume -1 >= a; then --1<=-a by REAL_1:50; then -a<= (-a)^2 by Th2; hence -a <= a^2 by SQUARE_1:61; end; theorem Th4: for a being real number st -1 > a holds -a < a^2 proof let a be real number;assume -1 > a; then --1< -a by REAL_1:50; then -a< (-a)^2 by SQUARE_1:76; hence -a < a^2 by SQUARE_1:61; end; theorem Th5: for a,b being real number st b^2<= a^2 & a>=0 holds -a<=b & b<=a proof let a,b be real number; assume A1:b^2<= a^2 & a>=0; now assume A2:-a>b or b>a; now per cases by A2; case -a>b; then --a<-b by REAL_1:50; then a^2<(-b)^2 by A1,SQUARE_1:78; hence contradiction by A1,SQUARE_1:61; case b>a; hence contradiction by A1,SQUARE_1:78; end; hence contradiction; end; hence -a<=b & b<=a; end; theorem Th6: for a,b being real number st b^2< a^2 & a>=0 holds -a<b & b<a proof let a,b be real number; assume A1:b^2< a^2 & a>=0; now assume A2:-a>=b or b>=a; now per cases by A2; case -a>=b; then --a<= -b by REAL_1:50; then a^2<=(-b)^2 by A1,SQUARE_1:77; hence contradiction by A1,SQUARE_1:61; case b>=a; hence contradiction by A1,SQUARE_1:77; end; hence contradiction; end; hence -a<b & b<a; end; theorem for a,b being real number st -a<=b & b<=a holds b^2<= a^2 proof let a,b be real number;assume A1: -a<=b & b<=a; per cases; suppose b>=0; hence b^2<= a^2 by A1,SQUARE_1:77; suppose b<0; then A2: -b>0 by REAL_1:66; --a>=-b by A1,REAL_1:50; then (-b)^2<= a^2 by A2,SQUARE_1:77; hence b^2<= a^2 by SQUARE_1:61; end; theorem Th8: for a,b being real number st -a<b & b<a holds b^2< a^2 proof let a,b be real number;assume A1: -a<b & b<a; per cases; suppose b>=0; hence b^2< a^2 by A1,SQUARE_1:78; suppose b<0; then A2: -b>0 by REAL_1:66; --a>-b by A1,REAL_1:50; then (-b)^2< a^2 by A2,SQUARE_1:78; hence b^2< a^2 by SQUARE_1:61; end; reserve T,T1,T2,S for non empty TopSpace; theorem Th9: :: BORSUK_2:1 for f being map of T1,S, g being map of T2,S,F1,F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1=[#] T1 & F2=[#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p be set st p in ([#] T1) /\ ([#] T2) holds f.p = g.p ) ex h being map of T,S st h = f+*g & h is continuous proof let f be map of T1,S, g be map of T2,S,F1,F2 being Subset of T; assume that A1: T1 is SubSpace of T & T2 is SubSpace of T and A2:F1=[#] T1 & F2=[#] T2 and A3: ([#] T1) \/ ([#] T2) = [#] T and A4: F1 is closed and A5: F2 is closed and A6: f is continuous and A7: g is continuous and A8: for p be set st p in ([#] T1) /\ ([#] T2) holds f.p = g.p; set h = f+*g; A9: dom f = the carrier of T1 by FUNCT_2:def 1 .= [#] T1 by PRE_TOPC:12; A10: dom g = the carrier of T2 by FUNCT_2:def 1 .= [#] T2 by PRE_TOPC:12; then A11: dom h = [#] T by A3,A9,FUNCT_4:def 1 .= the carrier of T by PRE_TOPC:12; rng f \/ rng g c= the carrier of S & rng h c= rng f \/ rng g by FUNCT_4:18; then rng h c= the carrier of S by XBOOLE_1:1; then h is Function of the carrier of T, the carrier of S by A11,FUNCT_2:4; then reconsider h as map of T,S; take h; thus h = f+*g; for P being Subset of S st P is closed holds h"P is closed proof let P be Subset of S; assume A12: P is closed; A13: h"P c= dom h & dom h = dom f \/ dom g by FUNCT_4:def 1,RELAT_1:167; then A14: h"P = h"P /\ ([#] T1 \/ [#] T2) by A9,A10,XBOOLE_1:28 .= (h"P /\ [#](T1)) \/ (h"P /\ [#](T2)) by XBOOLE_1:23; A15: for x being set st x in [#] T1 holds h.x = f.x proof let x be set such that A16: x in [#] T1; now per cases; suppose A17: x in [#] T2; then x in [#] T1 /\ [#] T2 by A16,XBOOLE_0:def 3; then f.x = g.x by A8; hence thesis by A10,A17,FUNCT_4:14; suppose not x in [#] T2; hence h.x = f.x by A10,FUNCT_4:12; end; hence thesis; end; now let x be set; thus x in h"P /\ [#] T1 implies x in f"P proof assume x in h"P /\ [#] T1; then x in h"P & x in dom f & x in [#] T1 by A9,XBOOLE_0:def 3; then h.x in P & x in dom f & f.x = h.x by A15,FUNCT_1:def 13; hence x in f"P by FUNCT_1:def 13; end; assume x in f"P; then x in dom f & f.x in P by FUNCT_1:def 13; then x in dom h & x in [#] T1 & h.x in P by A9,A13,A15,XBOOLE_0:def 2; then x in h"P & x in [#] T1 by FUNCT_1:def 13; hence x in h"P /\ [#] T1 by XBOOLE_0:def 3; end; then A18: h"P /\ [#] T1 = f"P by TARSKI:2; now let x be set; thus x in h"P /\ [#] T2 implies x in g"P proof assume x in h"P /\ [#] T2; then x in h"P & x in dom g & x in [#] T2 by A10,XBOOLE_0:def 3; then h.x in P & x in dom g & g.x = h.x by FUNCT_1:def 13,FUNCT_4:14; hence x in g"P by FUNCT_1:def 13; end; assume x in g"P; then x in dom g & g.x in P by FUNCT_1:def 13; then x in dom h & x in [#] T2 & h.x in P by A10,A13,FUNCT_4:14,XBOOLE_0:def 2; then x in h"P & x in [#] T2 by FUNCT_1:def 13; hence x in h"P /\ [#] T2 by XBOOLE_0:def 3; end; then A19: h"P = f"P \/ g"P by A14,A18,TARSKI:2; f"P c= the carrier of T1; then f"P c= [#] T1 & [#] T1 c= [#] T by A3,PRE_TOPC:12,XBOOLE_1:7; then f"P c= [#] T by XBOOLE_1:1; then f"P is Subset of T by PRE_TOPC:12; then reconsider P1 = f"P as Subset of T; g"P c= the carrier of T2; then g"P c= [#] T2 & [#] T2 c= [#] T by A3,PRE_TOPC:12,XBOOLE_1:7; then g"P c= [#] T by XBOOLE_1:1; then g"P is Subset of T by PRE_TOPC:12; then reconsider P2 = g"P as Subset of T; set P3 = f"P, P4 = g"P; A20:P3 is closed & P4 is closed by A6,A7,A12,PRE_TOPC:def 12; then consider F01 being Subset of T such that A21: F01 is closed & P3=F01 /\ [#]T1 by A1,PRE_TOPC:43; A22:P1 is closed by A2,A4,A21,TOPS_1:35; consider F02 being Subset of T such that A23: F02 is closed & P4=F02 /\ [#]T2 by A1,A20,PRE_TOPC:43; P2 is closed by A2,A5,A23,TOPS_1:35; hence h"P is closed by A19,A22,TOPS_1:36; end; hence thesis by PRE_TOPC:def 12; end; theorem Th10:for n being Nat,q2 being Point of Euclid n, q being Point of TOP-REAL n, r being real number st q=q2 holds Ball(q2,r) = {q3 where q3 is Point of TOP-REAL n: |.q-q3.|<r} proof let n be Nat,q2 be Point of (Euclid n), q be Point of TOP-REAL n,r be real number; assume A1:q=q2; A2:Ball(q2,r)= {q4 where q4 is Element of Euclid n: dist(q2,q4) < r} by METRIC_1:18; A3:{q4 where q4 is Element of Euclid n: dist(q2,q4) < r} c= {q3 where q3 is Point of TOP-REAL n: |.q-q3.|<r} proof let x be set;assume x in {q4 where q4 is Element of Euclid n: dist(q2,q4) < r}; then consider q4 being Element of Euclid n such that A4: q4=x & dist(q2,q4) < r; reconsider q44=q4 as Point of TOP-REAL n by TOPREAL3:13; dist(q2,q4)=|.q-q44.| by A1,JGRAPH_1:45; hence thesis by A4; end; {q3 where q3 is Point of TOP-REAL n: |.q-q3.|<r} c={q4 where q4 is Element of Euclid n: dist(q2,q4) < r} proof let x be set;assume x in {q3 where q3 is Point of TOP-REAL n: |.q-q3.|<r}; then consider q3 being Point of TOP-REAL n such that A5: x=q3 & |.q-q3.|<r; reconsider q34=q3 as Point of Euclid n by TOPREAL3:13; dist(q2,q34)=|.q-q3.| by A1,JGRAPH_1:45; hence thesis by A5; end; hence thesis by A2,A3,XBOOLE_0:def 10; end; theorem Th11: (0.REAL 2)`1=0 & (0.REAL 2)`2=0 by EUCLID:56,58; theorem Th12: 1.REAL 2 = <* 1, 1 *> proof reconsider f= (2 qua Nat |-> (1 qua Real)) as FinSequence of REAL by FINSEQ_2:77; thus 1.REAL 2=1*2 by JORDAN2C:def 8 .=f by JORDAN2C:def 7 .=<* 1 qua Real,1 qua Real *> by FINSEQ_2:75; end; theorem Th13: (1.REAL 2)`1=1 & (1.REAL 2)`2=1 proof 1.REAL 2=|[1,1]| by Th12,EUCLID:def 16; hence thesis by EUCLID:56; end; theorem Th14: dom proj1=the carrier of TOP-REAL 2 & dom proj1=REAL 2 proof thus dom proj1=the carrier of TOP-REAL 2 by FUNCT_2:def 1; hence thesis by EUCLID:25; end; theorem Th15: dom proj2=the carrier of TOP-REAL 2 & dom proj2=REAL 2 proof thus dom proj2=the carrier of TOP-REAL 2 by FUNCT_2:def 1; hence thesis by EUCLID:25; end; theorem proj1 is map of TOP-REAL2,R^1 by TOPMETR:24; theorem proj2 is map of TOP-REAL2,R^1 by TOPMETR:24; theorem Th18: for p being Point of TOP-REAL 2 holds p=|[proj1.p,proj2.p]| proof let p be Point of TOP-REAL 2; A1:p=|[p`1,p`2]| by EUCLID:57; p`1=proj1.p by PSCOMP_1:def 28; hence thesis by A1,PSCOMP_1:def 29; end; theorem Th19: for B being Subset of TOP-REAL 2 st B={0.REAL 2} holds B`<>{} & (the carrier of TOP-REAL 2)\B<>{} proof let B be Subset of TOP-REAL 2; assume A1:B={0.REAL 2}; now assume |[0,1]| in B; then |[0,1]|`2=0 by A1,Th11,TARSKI:def 1; hence contradiction by EUCLID:56; end; then |[0,1]| in (the carrier of TOP-REAL 2) \ B by XBOOLE_0:def 4; hence thesis by SUBSET_1:def 5; end; theorem Th20: :: BORSUK_1:def 2 for X,Y being non empty TopSpace,f being map of X,Y holds f is continuous iff for p being Point of X,V being Subset of Y st f.p in V & V is open holds ex W being Subset of X st p in W & W is open & f.:W c= V proof let X,Y be non empty TopSpace,f be map of X,Y; A1:dom f=the carrier of X by FUNCT_2:def 1; hereby assume A2: f is continuous; thus for p being Point of X,V being Subset of Y st f.p in V & V is open holds ex W being Subset of X st p in W & W is open & f.:W c= V proof let p be Point of X,V be Subset of Y; assume A3:f.p in V & V is open; then A4:f"V is open by A2,TOPS_2:55; A5:p in f"V by A1,A3,FUNCT_1:def 13; f.:(f"V) c= V by FUNCT_1:145; hence ex W being Subset of X st p in W & W is open & f.:W c= V by A4,A5; end; end; assume A6:for p being Point of X,V being Subset of Y st f.p in V & V is open holds ex W being Subset of X st p in W & W is open & f.:W c= V; for G being Subset of Y st G is open holds f"G is open proof let G be Subset of Y; assume A7:G is open; for z being set holds z in f"G iff ex Q being Subset of X st Q is open & Q c= f"G & z in Q proof let z be set; now assume A8:z in f"G; then A9: z in dom f & f.z in G by FUNCT_1:def 13; reconsider p=z as Point of X by A8; consider W being Subset of X such that A10: p in W & W is open & f.:W c= G by A6,A7,A9; A11: f"(f.:W) c= f"G by A10,RELAT_1:178; W c= f"(f.:W) by A1,FUNCT_1:146; then W c= f"G by A11,XBOOLE_1:1; hence ex Q being Subset of X st Q is open & Q c= f"G & z in Q by A10; end; hence thesis; end; hence f"G is open by TOPS_1:57; end; hence thesis by TOPS_2:55; end; theorem Th21: for p being Point of TOP-REAL 2, G being Subset of TOP-REAL 2 st G is open & p in G ex r being real number st r>0 & {q where q is Point of TOP-REAL 2: p`1-r<q`1 & q`1<p`1+r & p`2-r<q`2 & q`2<p`2+r} c= G proof let p be Point of TOP-REAL 2,G being Subset of TOP-REAL 2; assume A1: G is open & p in G; reconsider q2=p as Point of Euclid 2 by TOPREAL3:13; consider r being real number such that A2: r>0 & Ball(q2,r) c= G by A1,Lm1,TOPMETR:22; set s=r/sqrt(2); sqrt 2>0 by SQUARE_1:93; then A3: s>0 by A2,REAL_2:127; A4: Ball(q2,r)= {q3 where q3 is Point of TOP-REAL 2: |.p-q3.|<r} by Th10; {q where q is Point of TOP-REAL 2: p`1-s<q`1 & q`1<p`1+s & p`2-s<q`2 & q`2<p`2+s} c= Ball(q2,r) proof let x be set;assume x in {q where q is Point of TOP-REAL 2: p`1-s<q`1 & q`1<p`1+s & p`2-s<q`2 & q`2<p`2+s}; then consider q being Point of TOP-REAL 2 such that A5: q=x & ( p`1-s<q`1 & q`1<p`1+s & p`2-s<q`2 & q`2<p`2+s); A6:(|.p-q.|)^2=((p-q)`1)^2+((p-q)`2)^2 by JGRAPH_1:46; A7:(p-q)`1=p`1-q`1 & (p-q)`2=p`2-q`2 by TOPREAL3:8; p`1-s+s<q`1+s by A5,REAL_1:67; then p`1<q`1+s by XCMPLX_1:27; then p`1-q`1<q`1+s-q`1 by REAL_1:92; then A8: p`1-q`1<s by XCMPLX_1:26; p`1+s-s>q`1-s by A5,REAL_1:92; then p`1>q`1-s by XCMPLX_1:26; then p`1-q`1>q`1-s-q`1 by REAL_1:92; then p`1-q`1>q`1+-s-q`1 by XCMPLX_0:def 8; then p`1-q`1>-s by XCMPLX_1:26; then A9: (p`1-q`1)^2<s^2 by A8,Th8; p`2-s+s<q`2+s by A5,REAL_1:67; then p`2<q`2+s by XCMPLX_1:27; then p`2-q`2<q`2+s-q`2 by REAL_1:92; then A10: p`2-q`2<s by XCMPLX_1:26; p`2+s-s>q`2-s by A5,REAL_1:92; then p`2>q`2-s by XCMPLX_1:26; then p`2-q`2>q`2-s-q`2 by REAL_1:92; then p`2-q`2>q`2+-s-q`2 by XCMPLX_0:def 8; then p`2-q`2>-s by XCMPLX_1:26; then A11:(p`2-q`2)^2<s^2 by A10,Th8; s^2=r^2/(sqrt(2))^2 by SQUARE_1:69 .=r^2/2 by SQUARE_1:def 4; then s^2+s^2=r^2 by XCMPLX_1:66; then (|.p-q.|)^2<r^2 by A6,A7,A9,A11,REAL_1:67; then |.p-q.|<r by A2,Th6; hence x in Ball(q2,r) by A4,A5; end; then {q where q is Point of TOP-REAL 2: p`1-s<q`1 & q`1<p`1+s & p`2-s<q`2 & q`2<p`2+s} c= G by A2,XBOOLE_1:1; hence thesis by A3; end; theorem Th22: for X,Y,Z being non empty TopSpace, B being Subset of Y, C being Subset of Z, f being map of X,Y, h being map of Y|B,Z|C st f is continuous & h is continuous & rng f c= B & B<>{} & C<>{} holds ex g being map of X,Z st g is continuous & g=h*f proof let X,Y,Z be non empty TopSpace, B be Subset of Y, C be Subset of Z, f be map of X,Y , h be map of Y|B,Z|C; assume A1:f is continuous & h is continuous & rng f c= B & B<>{} & C<>{}; then reconsider V=B as non empty Subset of Y; reconsider F=C as non empty Subset of Z by A1; A2:Z|F is non empty; A3:Y|V is non empty; A4:the carrier of Y|B=[#](Y|B) by PRE_TOPC:12 .=B by PRE_TOPC:def 10; the carrier of X=dom f by FUNCT_2:def 1; then f is Function of the carrier of X,the carrier of Y|B by A1,A4,FUNCT_2:4; then reconsider u=f as map of X,Y|B; reconsider G=Z|C as non empty TopSpace by A2; reconsider H=Y|B as non empty TopSpace by A3; reconsider k=u as map of X,H; A5:u is continuous by A1,TOPMETR:9; reconsider j=h as map of H,G; A6:j*k is map of X,G; then reconsider w=h*k as map of X,G; A7:w is continuous by A1,A5,TOPS_2:58; the carrier of (Z|C)=[#](Z|C) by PRE_TOPC:12 .=C by PRE_TOPC:def 10; then h*u is Function of the carrier of X,the carrier of Z by A6,FUNCT_2:9; then reconsider v=h*u as map of X,Z; v is continuous by A7,TOPMETR:7; hence thesis; end; reserve p,q for Point of TOP-REAL 2; definition func Out_In_Sq -> Function of (the carrier of TOP-REAL 2)\{0.REAL 2}, (the carrier of TOP-REAL 2)\{0.REAL 2} means :Def1: for p being Point of TOP-REAL 2 st p<>0.REAL 2 holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies it.p=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies it.p=|[p`1/p`2/p`2,1/p`2]|); existence proof reconsider BP= (the carrier of TOP-REAL 2)\{0.REAL 2} as non empty set by Th19; A1:BP c= the carrier of TOP-REAL 2 by XBOOLE_1:36; defpred P[set,set] means (for p being Point of TOP-REAL 2 st p=$1 holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies $2=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies $2=|[p`1/p`2/p`2,1/p`2]|)); A2:for x being Element of BP ex y being Element of BP st P[x,y] proof let x be Element of BP; reconsider q=x as Point of TOP-REAL 2 by A1,TARSKI:def 3; now per cases; case A3:(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1); now assume |[1/q`1,q`2/q`1/q`1]| in {0.REAL 2}; then 0.REAL 2= |[1/q`1,q`2/q`1/q`1]| by TARSKI:def 1; then 0=1/q`1 & 0=q`2/q`1/q`1 by Th11,EUCLID:56; then A4:0=1/q`1*q`1; now per cases; case A5:q`1=0; then q`2=0 by A3; then q=0.REAL 2 by A5,EUCLID:57,58; then q in {0.REAL 2} by TARSKI:def 1; hence contradiction by XBOOLE_0:def 4; case q`1<>0; hence contradiction by A4,XCMPLX_1:88; end; hence contradiction; end; then reconsider r= |[1/q`1,q`2/q`1/q`1]| as Element of BP by XBOOLE_0:def 4; for p being Point of TOP-REAL 2 st p=x holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies r=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies r=|[p`1/p`2/p`2,1/p`2]|) by A3; hence ex y being Element of BP st (for p being Point of TOP-REAL 2 st p=x holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies y=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies y=|[p`1/p`2/p`2,1/p`2]|)); case A6:not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1); now assume |[q`1/q`2/q`2,1/q`2]| in {0.REAL 2}; then 0.REAL 2= |[q`1/q`2/q`2,1/q`2]| by TARSKI:def 1; then (0.REAL 2)`2=1/q`2 & (0.REAL 2)`1= q`1/q`2/q`2 by EUCLID:56; then A7:0=1/q`2*q`2 by Th11; now per cases; case q`2=0; then not (0<=q`1 & -0<=q`1 or 0>=q`1 & 0<=-q`1) by A6,REAL_2:110; then not (0<=q`1 & 0<=q`1 or 0>=q`1 & q`1<= -0) by REAL_2:110; hence contradiction; case q`2<>0; hence contradiction by A7,XCMPLX_1:88; end; hence contradiction; end; then reconsider r= |[q`1/q`2/q`2,1/q`2]| as Element of BP by XBOOLE_0:def 4; for p being Point of TOP-REAL 2 st p=x holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies r=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies r=|[p`1/p`2/p`2,1/p`2]|) by A6; hence thesis; end; hence thesis; end; ex h being Function of BP, BP st for x being Element of BP holds P[x,h.x] from FuncExD(A2); then consider h being Function of BP, BP such that A8: for x being Element of BP holds for p being Point of TOP-REAL 2 st p=x holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies h.x=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies h.x=|[p`1/p`2/p`2,1/p`2]|); for p being Point of TOP-REAL 2 st p<>0.REAL 2 holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies h.p=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies h.p=|[p`1/p`2/p`2,1/p`2]|) proof let p be Point of TOP-REAL 2;assume p<>0.REAL 2; then not p in {0.REAL 2} by TARSKI:def 1; then p in (the carrier of TOP-REAL 2)\{0.REAL 2} by XBOOLE_0:def 4; hence thesis by A8; end; hence thesis; end; uniqueness proof let h1,h2 be Function of (the carrier of TOP-REAL 2)\{0.REAL 2}, (the carrier of TOP-REAL 2)\{0.REAL 2}; assume A9: ( for p being Point of TOP-REAL 2 st p<>0.REAL 2 holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies h1.p=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies h1.p=|[p`1/p`2/p`2,1/p`2]|))& ( for p being Point of TOP-REAL 2 st p<>0.REAL 2 holds ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)implies h2.p=|[1/p`1,p`2/p`1/p`1]|) & (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies h2.p=|[p`1/p`2/p`2,1/p`2]|)); reconsider BP= (the carrier of TOP-REAL 2)\{0.REAL 2} as non empty set by Th19; A10:BP c= the carrier of TOP-REAL 2 by XBOOLE_1:36; for x being set st x in (the carrier of TOP-REAL 2)\{0.REAL 2} holds h1.x=h2.x proof let x be set; assume A11: x in (the carrier of TOP-REAL 2)\{0.REAL 2}; then reconsider q=x as Point of TOP-REAL 2 by A10; not q in {0.REAL 2} by A11,XBOOLE_0:def 4; then A12:q<>0.REAL 2 by TARSKI:def 1; now per cases; case A13:(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1); then h1.q=|[1/q`1,q`2/q`1/q`1]| by A9,A12; hence h1.x=h2.x by A9,A12,A13; case A14:not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1); then h1.q=|[q`1/q`2/q`2,1/q`2]| by A9,A12; hence h1.x=h2.x by A9,A12,A14; end; hence h1.x=h2.x; end; hence h1=h2 by FUNCT_2:18; end; end; theorem Th23: for p being Point of TOP-REAL 2 st not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) holds p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2 proof let p being Point of TOP-REAL 2; assume A1:not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1); A2:-p`1<p`2 implies --p`1>-p`2 by REAL_1:50; -p`1>p`2 implies --p`1<-p`2 by REAL_1:50; hence thesis by A1,A2; end; theorem Th24: for p being Point of TOP-REAL 2 st p<>0.REAL 2 holds ((p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2)implies Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]|) & (not (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) implies Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]|) proof let p be Point of TOP-REAL 2;assume A1: p<>0.REAL 2; hereby assume A2:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2); now per cases by A2; case A3:p`1<=p`2 & -p`2<=p`1; now assume A4:p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1; A5: now per cases by A4; case p`2<=p`1 & -p`1<=p`2; hence p`1=p`2 or p`1=-p`2 by A3,AXIOMS:21; case p`2>=p`1 & p`2<=-p`1; then -p`2>=--p`1 by REAL_1:50; hence p`1=p`2 or p`1=-p`2 by A3,AXIOMS:21; end; now per cases by A5; case A6:p`1=p`2; now assume p`1=0; hence contradiction by A1,A6,EUCLID:57,58; end; then p`1/p`2/p`2=1/p`1 by A6,XCMPLX_1:60; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A4,A6,Def1; case A7:p`1=-p`2; A8:now assume A9:p`1=0; then p`2=-0 by A7; hence contradiction by A1,A9,EUCLID:57,58; end; A10:-p`1=p`2 by A7; A11:p`2<>0 by A7,A8; A12:p`1/p`2/p`2=(-(p`2/p`2))/p`2 by A7,XCMPLX_1:188 .=(-1)/p`2 by A11,XCMPLX_1:60 .= 1/p`1 by A7,XCMPLX_1:193; 1/p`2= -(1/p`1) by A10,XCMPLX_1:189 .=-(p`2/p`1/(-p`1)) by A7,A12,XCMPLX_1:193 .=--(p`2/p`1/p`1) by XCMPLX_1:189.=p`2/p`1/p`1; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A4,A12,Def1; end; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]|; end; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,Def1; case A13:p`1>=p`2 & p`1<=-p`2; now assume A14:p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1; A15: now per cases by A14; case p`2<=p`1 & -p`1<=p`2; then --p`1>=-p`2 by REAL_1:50; hence p`1=p`2 or p`1=-p`2 by A13,AXIOMS:21; case p`2>=p`1 & p`2<=-p`1; hence p`1=p`2 or p`1=-p`2 by A13,AXIOMS:21; end; now per cases by A15; case A16:p`1=p`2; then p`1 <> 0 by A1,EUCLID:57,58; then p`1/p`2/p`2=1/p`1 by A16,XCMPLX_1:60; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A14,A16,Def1; case A17:p`1=-p`2; A18:now assume A19:p`1=0; then p`2=-0 by A17; hence contradiction by A1,A19,EUCLID:57,58; end; A20:-p`1=p`2 by A17; A21:p`2<>0 by A17,A18; A22:p`1/p`2/p`2 =(-(p`2/p`2))/p`2 by A17,XCMPLX_1:188 .=(-1)/p`2 by A21,XCMPLX_1:60 .= 1/p`1 by A17,XCMPLX_1:193; then 1/p`2=-(p`1/p`2/p`2) by A20,XCMPLX_1:189 .=-(p`2/p`1/(-p`1)) by A17,XCMPLX_1:192 .=--(p`2/p`1/p`1) by XCMPLX_1:189.=p`2/p`1/p`1; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,A14,A22,Def1; end; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]|; end; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A1,Def1; end; hence Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]|; end; hereby assume A23:not(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2); A24:-p`2<p`1 implies --p`2>-p`1 by REAL_1:50; -p`2>p`1 implies --p`2<-p`1 by REAL_1:50; hence Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by A1,A23,A24,Def1; end; end; theorem Th25: for D being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL 2)|D st K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} holds rng (Out_In_Sq|K0) c= the carrier of ((TOP-REAL 2)|D)|K0 proof let D be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|D; assume A1: K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2}; A2:the carrier of ((TOP-REAL 2)|D) =[#]((TOP-REAL 2)|D) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; let y be set;assume y in rng (Out_In_Sq|K0); then consider x being set such that A3:x in dom (Out_In_Sq|K0) & y=(Out_In_Sq|K0).x by FUNCT_1:def 5; A4:x in (dom Out_In_Sq) /\ K0 by A3,FUNCT_1:68; then A5:x in K0 by XBOOLE_0:def 3; A6: K0 c= the carrier of TOP-REAL 2 by A2,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A5; A7:Out_In_Sq.p=y by A3,A5,FUNCT_1:72; consider px being Point of TOP-REAL 2 such that A8: x=px & (px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1) & px<>0.REAL 2 by A1,A5; reconsider K00=K0 as Subset of TOP-REAL 2 by A6; K00=[#]((TOP-REAL 2)|K00) by PRE_TOPC:def 10 .=the carrier of ((TOP-REAL 2)|K00) by PRE_TOPC:12; then A9:p in the carrier of ((TOP-REAL 2)|K00) by A4,XBOOLE_0:def 3; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K00 holds q`1<>0 proof let q be Point of TOP-REAL 2; assume A10:q in the carrier of (TOP-REAL 2)|K00; the carrier of (TOP-REAL 2)|K00=[#]((TOP-REAL 2)|K00) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A11: q=p3 & ((p3`2<=p3`1 & -p3`1<=p3`2 or p3`2>=p3`1 & p3`2<=-p3`1) & p3<>0.REAL 2) by A1,A10; now assume A12:q`1=0; then q`2=0 by A11; hence contradiction by A11,A12,EUCLID:57,58; end; hence q`1<>0; end; then A13:p`1<>0 by A9; set p9=|[1/p`1,p`2/p`1/p`1]|; A14:p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:56; A15:now assume p9=0.REAL 2; then 0 *p`1=1/p`1*p`1 by A14,EUCLID:56,58; hence contradiction by A13,XCMPLX_1:88; end; A16:Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by A8,Def1; now per cases; case A17: p`1>=0; then p`2/p`1<=p`1/p`1 & (-1 *p`1)/p`1<=p`2/p`1 or p`2>=p`1 & p`2<=-1 *p `1 by A8,A13,REAL_1:73; then p`2/p`1<=1 & (-1)*p`1/p`1<=p`2/p`1 or p`2>=p`1 & p`2<=-1 *p`1 by A13,XCMPLX_1:60,175; then p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=p`1/p`1 & p`2<=-1 *p`1 by A13,A17,REAL_1:73,XCMPLX_1:90; then p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=1 & p`2<=(-1)*p`1 by A13,XCMPLX_1:60,175; then p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=1 & p`2/p`1<=(-1)*p`1/p`1 by A13,A17,REAL_1:73; then A18:p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=1 & p`2/p`1<=-1 by A13,XCMPLX_1:90; then (-1)/p`1<= p`2/p`1/p`1 by A13,A17,AXIOMS:22,REAL_1:73; then A19:p`2/p`1/p`1 <=1/p`1 & -(1/p`1)<= p`2/p`1/p`1 or p`2/p`1/p`1 >=1/p`1 & p`2/p`1/p`1<= -(1/p`1) by A13,A17,A18,AXIOMS:22,REAL_1:73,XCMPLX_1:188; p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:56; hence y in K0 by A1,A7,A15,A16,A19; case A20:p`1<0; then p`2<=p`1 & (-1 *p`1)<=p`2 or p`2/p`1<=p`1/p`1 & p`2/p`1>=(-1 *p`1)/ p`1 by A8,REAL_1:74; then p`2<=p`1 & (-1 *p`1)<=p`2 or p`2/p`1<=1 & p`2/p`1>=(-1)*p`1/p`1 by A20,XCMPLX_1:60,175; then A21: p`2/p`1>=p`1/p`1 & (-1 *p`1)<=p`2 or p`2/p`1<=1 & p`2/p`1>=-1 by A20,REAL_1:74,XCMPLX_1:90; then p`2/p`1>=1 & (-1)*p`1<=p`2 or p`2/p`1<=1 & p`2/p`1>=-1 by A20,XCMPLX_1:60,175; then p`2/p`1>=1 & (-1)*p`1/p`1>=p`2/p`1 or p`2/p`1<=1 & p`2/p`1>=-1 by A20,REAL_1:74; then A22:p`2/p`1>=1 & -1>=p`2/p`1 or p`2/p`1<=1 & p`2/p`1>=-1 by A20,XCMPLX_1:90; not (p`2/p`1>=1 & p`2/p`1<=-1) by AXIOMS:22; then (-1)/p`1>= p`2/p`1/p`1 by A20,A21,REAL_1:74,XCMPLX_1:60; then A23:p`2/p`1/p`1 <=1/p`1 & -(1/p`1)<= p`2/p`1/p`1 or p`2/p`1/p`1 >=1/p`1 & p`2/p`1/p`1<= -(1/p`1) by A20,A22,AXIOMS:22,REAL_1:74,XCMPLX_1:188; p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:56; hence y in K0 by A1,A7,A15,A16,A23; end; then y in [#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 10; hence thesis; end; theorem Th26: for D being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL 2)|D st K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} holds rng (Out_In_Sq|K0) c= the carrier of ((TOP-REAL 2)|D)|K0 proof let D be Subset of TOP-REAL 2, K0 be Subset of (TOP-REAL 2)|D; assume A1: K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2}; A2:the carrier of ((TOP-REAL 2)|D) =[#]((TOP-REAL 2)|D) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; let y be set;assume y in rng (Out_In_Sq|K0); then consider x being set such that A3:x in dom (Out_In_Sq|K0) & y=(Out_In_Sq|K0).x by FUNCT_1:def 5; x in (dom Out_In_Sq) /\ K0 by A3,FUNCT_1:68; then A4:x in K0 by XBOOLE_0:def 3; A5: K0 c= the carrier of TOP-REAL 2 by A2,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A4; A6:Out_In_Sq.p=y by A3,A4,FUNCT_1:72; consider px being Point of TOP-REAL 2 such that A7: x=px & (px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2) & px<>0.REAL 2 by A1,A4; reconsider K00=K0 as Subset of TOP-REAL 2 by A5; A8:K00=[#]((TOP-REAL 2)|K00) by PRE_TOPC:def 10 .=the carrier of ((TOP-REAL 2)|K00) by PRE_TOPC:12; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K00 holds q`2<>0 proof let q be Point of TOP-REAL 2; assume A9:q in the carrier of (TOP-REAL 2)|K00; the carrier of (TOP-REAL 2)|K00=[#]((TOP-REAL 2)|K00) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A10: q=p3 & ((p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2) & p3<>0.REAL 2) by A1,A9; now assume A11:q`2=0; then q`1=0 by A10; hence contradiction by A10,A11,EUCLID:57,58; end; hence q`2<>0; end; then A12:p`2<>0 by A4,A8; set p9=|[p`1/p`2/p`2,1/p`2]|; A13:p9`2=1/p`2 & p9`1=p`1/p`2/p`2 by EUCLID:56; A14:now assume p9=0.REAL 2; then 0 *p`2=1/p`2*p`2 by A13,EUCLID:56,58; hence contradiction by A12,XCMPLX_1:88; end; A15:Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A7,Th24; now per cases; case A16: p`2>=0; then p`1/p`2<=p`2/p`2 & (-1 *p`2)/p`2<=p`1/p`2 or p`1>=p`2 & p`1<=-1 *p `2 by A7,A12,REAL_1:73; then p`1/p`2<=1 & (-1)*p`2/p`2<=p`1/p`2 or p`1>=p`2 & p`1<=-1 *p`2 by A12,XCMPLX_1:60,175; then p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=p`2/p`2 & p`1<=-1 *p`2 by A12,A16,REAL_1:73,XCMPLX_1:90; then p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=1 & p`1<=(-1)*p`2 by A12,XCMPLX_1:60,175; then p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=1 & p`1/p`2<=(-1)*p`2/p`2 by A12,A16,REAL_1:73; then A17:p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=1 & p`1/p`2<=-1 by A12,XCMPLX_1:90; then (-1)/p`2<= p`1/p`2/p`2 by A12,A16,AXIOMS:22,REAL_1:73; then A18:p`1/p`2/p`2 <=1/p`2 & -(1/p`2)<= p`1/p`2/p`2 or p`1/p`2/p`2 >=1/p`2 & p`1/p`2/p`2<= -(1/p`2) by A12,A16,A17,AXIOMS:22,REAL_1:73,XCMPLX_1:188; p9`2=1/p`2 & p9`1=p`1/p`2/p`2 by EUCLID:56; hence y in K0 by A1,A6,A14,A15,A18; case A19:p`2<0; then p`1<=p`2 & (-1 *p`2)<=p`1 or p`1/p`2<=p`2/p`2 & p`1/p`2>=(-1 *p`2)/ p`2 by A7,REAL_1:74; then p`1<=p`2 & (-1 *p`2)<=p`1 or p`1/p`2<=1 & p`1/p`2>=(-1)*p`2/p`2 by A19,XCMPLX_1:60,175; then p`1/p`2>=p`2/p`2 & (-1 *p`2)<=p`1 or p`1/p`2<=1 & p`1/p`2>=-1 by A19,REAL_1:74,XCMPLX_1:90; then p`1/p`2>=1 & (-1)*p`2<=p`1 or p`1/p`2<=1 & p`1/p`2>=-1 by A19,XCMPLX_1:60,175; then p`1/p`2>=1 & (-1)*p`2/p`2>=p`1/p`2 or p`1/p`2<=1 & p`1/p`2>=-1 by A19,REAL_1:74; then A20:p`1/p`2>=1 & -1>=p`1/p`2 or p`1/p`2<=1 & p`1/p`2>=-1 by A19,XCMPLX_1:90; then (-1)/p`2>= p`1/p`2/p`2 by A19,AXIOMS:22,REAL_1:74; then A21:p`1/p`2/p`2 <=1/p`2 & -(1/p`2)<= p`1/p`2/p`2 or p`1/p`2/p`2 >=1/p`2 & p`1/p`2/p`2<= -(1/p`2) by A19,A20,AXIOMS:22,REAL_1:74,XCMPLX_1:188; p9`2=1/p`2 & p9`1=p`1/p`2/p`2 by EUCLID:56; hence y in K0 by A1,A6,A14,A15,A21; end; then y in [#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 10; hence thesis; end; theorem Th27:for K0a being set,D being non empty Subset of TOP-REAL 2 st K0a={p where p is Point of TOP-REAL 2: (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} & D`={0.REAL 2} holds K0a is non empty Subset of (TOP-REAL 2)|D & K0a is non empty Subset of TOP-REAL 2 proof let K0a be set,D be non empty Subset of TOP-REAL 2; assume A1: K0a={p where p is Point of TOP-REAL 2: (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} & D`={0.REAL 2}; ((1.REAL 2)`2<=(1.REAL 2)`1 & -(1.REAL 2)`1<=(1.REAL 2)`2 or (1.REAL 2)`2>=(1.REAL 2)`1 & (1.REAL 2)`2<=-(1.REAL 2)`1) & (1.REAL 2)<>0.REAL 2 by Th13,REVROT_1:19; then A2:1.REAL 2 in K0a by A1; A3:the carrier of (TOP-REAL 2)|D=[#]((TOP-REAL 2)|D) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; A4:K0a c= D proof let x be set;assume x in K0a; then consider p8 being Point of TOP-REAL 2 such that A5: x=p8 & ( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2) by A1; A6:D=D`` .=(the carrier of TOP-REAL 2) \ {0.REAL 2} by A1,SUBSET_1:def 5; not x in {0.REAL 2} by A5,TARSKI:def 1; hence x in D by A5,A6,XBOOLE_0:def 4; end; hence K0a is non empty Subset of ((TOP-REAL 2)|D) by A2,A3; K0a c= the carrier of TOP-REAL 2 by A4,XBOOLE_1:1; hence thesis by A2; end; theorem Th28:for K0a being set,D being non empty Subset of TOP-REAL 2 st K0a={p where p is Point of TOP-REAL 2: (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} & D`={0.REAL 2} holds K0a is non empty Subset of (TOP-REAL 2)|D & K0a is non empty Subset of TOP-REAL 2 proof let K0a be set,D be non empty Subset of TOP-REAL 2; assume A1: K0a={p where p is Point of TOP-REAL 2: (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} & D`={0.REAL 2}; ((1.REAL 2)`1<=(1.REAL 2)`2 & -(1.REAL 2)`2<=(1.REAL 2)`1 or (1.REAL 2)`1>=(1.REAL 2)`2 & (1.REAL 2)`1<=-(1.REAL 2)`2) & (1.REAL 2)<>0.REAL 2 by Th13,REVROT_1:19; then A2:1.REAL 2 in K0a by A1; A3:the carrier of (TOP-REAL 2)|D=[#]((TOP-REAL 2)|D) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; A4:K0a c= D proof let x be set;assume x in K0a; then consider p8 being Point of TOP-REAL 2 such that A5: x=p8 & ( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2) by A1; A6:D=D`` .=(the carrier of TOP-REAL 2) \ {0.REAL 2} by A1,SUBSET_1:def 5; not x in {0.REAL 2} by A5,TARSKI:def 1; hence x in D by A5,A6,XBOOLE_0:def 4; end; hence K0a is non empty Subset of ((TOP-REAL 2)|D) by A2,A3; K0a c= the carrier of TOP-REAL 2 by A4,XBOOLE_1:1; hence thesis by A2; end; theorem Th29: for X being non empty TopSpace, f1,f2 being map of X,R^1 st f1 is continuous & f2 is continuous holds ex g being map of X,R^1 st (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g.p=r1+r2) & g is continuous proof let X being non empty TopSpace,f1,f2 be map of X,R^1; assume A1: f1 is continuous & f2 is continuous; defpred P[set,set] means (for r1,r2 being real number st f1.$1=r1 & f2.$1=r2 holds $2=r1+r2); A2:for x being Element of X ex y being Element of REAL st P[x,y] proof let x be Element of X; reconsider r1=f1.x as Real by TOPMETR:24; reconsider r2=f2.x as Real by TOPMETR:24; set r3=r1+r2; for r1,r2 being real number st f1.x=r1 & f2.x=r2 holds r3=r1+r2; hence ex y being Element of REAL st (for r1,r2 being real number st f1.x=r1 & f2.x=r2 holds y=r1+r2); end; ex f being Function of the carrier of X,REAL st for x being Element of X holds P[x,f.x] from FuncExD(A2); then consider f being Function of the carrier of X,REAL such that A3: for x being Element of X holds (for r1,r2 being real number st f1.x=r1 & f2.x=r2 holds f.x=r1+r2); reconsider g0=f as map of X,R^1 by TOPMETR:24; A4: for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g0.p=r1+r2 by A3; for p being Point of X,V being Subset of R^1 st g0.p in V & V is open holds ex W being Subset of X st p in W & W is open & g0.:W c= V proof let p be Point of X,V be Subset of R^1; assume A5:g0.p in V & V is open; reconsider r=g0.p as Real by TOPMETR:24; consider r0 being Real such that A6: r0>0 & ].r-r0,r+r0.[ c= V by A5,FRECHET:8; reconsider r1=f1.p as Real by TOPMETR:24; reconsider G1=].r1-r0/2,r1+r0/2.[ as Subset of R^1 by TOPMETR:24; r0/2>0 by A6,SEQ_2:3; then A7:r1<r1+r0/2 by REAL_1:69; then r1-r0/2<r1 by REAL_1:84; then A8:f1.p in G1 by A7,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A9: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A8,Th20; reconsider r2=f2.p as Real by TOPMETR:24; reconsider G2=].r2-r0/2,r2+r0/2.[ as Subset of R^1 by TOPMETR:24; r0/2>0 by A6,SEQ_2:3; then A10:r2<r2+r0/2 by REAL_1:69; then r2-r0/2<r2 by REAL_1:84; then A11:f2.p in G2 by A10,JORDAN6:45; G2 is open by JORDAN6:46; then consider W2 being Subset of X such that A12: p in W2 & W2 is open & f2.:W2 c= G2 by A1,A11,Th20; set W=W1 /\ W2; A13:W is open by A9,A12,TOPS_1:38; A14:p in W by A9,A12,XBOOLE_0:def 3; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A15: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; A16:z in W1 by A15,XBOOLE_0:def 3; reconsider pz=z as Point of X by A15; A17:pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A18:f1.pz in f1.:W1 by A16,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A19:z in W2 by A15,XBOOLE_0:def 3; pz in dom f2 by A17,FUNCT_2:def 1; then A20:f2.pz in f2.:W2 by A19,FUNCT_1:def 12; reconsider aa2=f2.pz as Real by TOPMETR:24; A21:x=aa1+aa2 by A3,A15; then reconsider rx=x as Real; A22:r1-r0/2<aa1 & aa1<r1+r0/2 by A9,A18,JORDAN6:45; A23:r2-r0/2<aa2 & aa2<r2+r0/2 by A12,A20,JORDAN6:45; then aa1+aa2<r1+r0/2+(r2+r0/2) by A22,REAL_1:67; then aa1+aa2<r1+r0/2+r2+r0/2 by XCMPLX_1:1; then aa1+aa2<r1+r2+r0/2+r0/2 by XCMPLX_1:1; then aa1+aa2<r1+r2+(r0/2+r0/2) by XCMPLX_1:1; then aa1+aa2<r1+r2+r0 by XCMPLX_1:66; then A24:rx<r+r0 by A3,A21; r1-r0/2+(r2-r0/2)<aa1+aa2 by A22,A23,REAL_1:67; then r1-r0/2+r2-r0/2<aa1+aa2 by XCMPLX_1:29; then r1+r2-r0/2-r0/2<aa1+aa2 by XCMPLX_1:29; then r1+r2-(r0/2+r0/2)<aa1+aa2 by XCMPLX_1:36; then r1+r2-r0<aa1+aa2 by XCMPLX_1:66; then r-r0<aa1+aa2 by A3; hence x in ].r-r0,r+r0.[ by A21,A24,JORDAN6:45; end; then g0.:W c= V by A6,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A13,A14; end; then g0 is continuous by Th20; hence thesis by A4; end; theorem for X being non empty TopSpace, a being real number holds ex g being map of X,R^1 st (for p being Point of X holds g.p=a) & g is continuous proof let X be non empty TopSpace,a be real number; deffunc F(set)=a; ex g being Function st dom g=the carrier of X & for x being set st x in the carrier of X holds g.x=F(x) from Lambda; then consider g1 being Function such that A1: dom g1=the carrier of X & for x being set st x in the carrier of X holds g1.x=a; rng g1 c= the carrier of R^1 proof let y be set;assume y in rng g1; then consider x being set such that A2: x in dom g1 & y=g1.x by FUNCT_1:def 5; a in REAL by XREAL_0:def 1; hence y in the carrier of R^1 by A1,A2,TOPMETR:24; end; then g1 is Function of the carrier of X,the carrier of R^1 by A1,FUNCT_2:4; then reconsider g0=g1 as map of X,R^1; A3:for p being Point of X holds g1.p=a by A1; for p being Point of X,V being Subset of R^1 st g0.p in V & V is open holds ex W being Subset of X st p in W & W is open & g0.:W c= V proof let p be Point of X,V be Subset of R^1; assume A4:g0.p in V & V is open; set f1=g0; set G1=V; A5:[#]X is open by TOPS_1:53; A6:f1.: [#]X c= G1 proof let y be set;assume y in f1.: [#]X; then consider x being set such that A7:x in dom f1 & x in [#]X & y=f1.x by FUNCT_1:def 12; y=a by A1,A7; hence y in G1 by A1,A4; end; p in the carrier of X; then p in [#]X by PRE_TOPC:12; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A5,A6; end; then g0 is continuous by Th20; hence thesis by A3; end; theorem Th31: for X being non empty TopSpace, f1,f2 being map of X,R^1 st f1 is continuous & f2 is continuous holds ex g being map of X,R^1 st (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g.p=r1-r2) & g is continuous proof let X being non empty TopSpace,f1,f2 be map of X,R^1; assume A1: f1 is continuous & f2 is continuous; defpred P[set,set] means (for r1,r2 being real number st f1.$1=r1 & f2.$1=r2 holds $2=r1-r2); A2:for x being Element of X ex y being Element of REAL st P[x,y] proof let x be Element of X; reconsider r1=f1.x as Real by TOPMETR:24; reconsider r2=f2.x as Real by TOPMETR:24; set r3=r1-r2; for r1,r2 being real number st f1.x=r1 & f2.x=r2 holds r3=r1-r2; hence ex y being Element of REAL st (for r1,r2 being real number st f1.x=r1 & f2.x=r2 holds y=r1-r2); end; ex f being Function of the carrier of X,REAL st for x being Element of X holds P[x,f.x] from FuncExD(A2); then consider f being Function of the carrier of X,REAL such that A3: for x being Element of X holds (for r1,r2 being real number st f1.x=r1 & f2.x=r2 holds f.x=r1-r2); reconsider g0=f as map of X,R^1 by TOPMETR:24; A4: for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g0.p=r1-r2 by A3; for p being Point of X,V being Subset of R^1 st g0.p in V & V is open holds ex W being Subset of X st p in W & W is open & g0.:W c= V proof let p be Point of X,V be Subset of R^1; assume A5:g0.p in V & V is open; reconsider r=g0.p as Real by TOPMETR:24; consider r0 being Real such that A6: r0>0 & ].r-r0,r+r0.[ c= V by A5,FRECHET:8; reconsider r1=f1.p as Real by TOPMETR:24; reconsider G1=].r1-r0/2,r1+r0/2.[ as Subset of R^1 by TOPMETR:24; r0/2>0 by A6,SEQ_2:3; then A7:r1<r1+r0/2 by REAL_1:69; then r1-r0/2<r1 by REAL_1:84; then A8:f1.p in G1 by A7,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A9: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A8,Th20; reconsider r2=f2.p as Real by TOPMETR:24; reconsider G2=].r2-r0/2,r2+r0/2.[ as Subset of R^1 by TOPMETR:24; r0/2>0 by A6,SEQ_2:3; then A10:r2<r2+r0/2 by REAL_1:69; then r2-r0/2<r2 by REAL_1:84; then A11:f2.p in G2 by A10,JORDAN6:45; G2 is open by JORDAN6:46; then consider W2 being Subset of X such that A12: p in W2 & W2 is open & f2.:W2 c= G2 by A1,A11,Th20; set W=W1 /\ W2; A13:W is open by A9,A12,TOPS_1:38; A14:p in W by A9,A12,XBOOLE_0:def 3; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A15: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; A16:z in W1 by A15,XBOOLE_0:def 3; reconsider pz=z as Point of X by A15; A17:pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A18:f1.pz in f1.:W1 by A16,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A19:z in W2 by A15,XBOOLE_0:def 3; pz in dom f2 by A17,FUNCT_2:def 1; then A20:f2.pz in f2.:W2 by A19,FUNCT_1:def 12; reconsider aa2=f2.pz as Real by TOPMETR:24; A21:x=aa1-aa2 by A3,A15; then reconsider rx=x as Real; A22:r1-r0/2<aa1 & aa1<r1+r0/2 by A9,A18,JORDAN6:45; A23:r2-r0/2<aa2 & aa2<r2+r0/2 by A12,A20,JORDAN6:45; then aa1-aa2<r1+r0/2-(r2-r0/2) by A22,REAL_1:92; then aa1-aa2<r1+r0/2-r2+r0/2 by XCMPLX_1:37; then aa1-aa2<r1-r2+r0/2+r0/2 by XCMPLX_1:29; then aa1-aa2<r1-r2+(r0/2+r0/2) by XCMPLX_1:1; then aa1-aa2<r1-r2+r0 by XCMPLX_1:66; then A24:rx<r+r0 by A3,A21; r1-r0/2-(r2+r0/2)<aa1-aa2 by A22,A23,REAL_1:92; then r1-r0/2-r2-r0/2<aa1-aa2 by XCMPLX_1:36; then r1-r2-r0/2-r0/2<aa1-aa2 by XCMPLX_1:21; then r1-r2-(r0/2+r0/2)<aa1-aa2 by XCMPLX_1:36; then r1-r2-r0<aa1-aa2 by XCMPLX_1:66; then r-r0<aa1-aa2 by A3; hence x in ].r-r0,r+r0.[ by A21,A24,JORDAN6:45; end; then g0.:W c= V by A6,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A13,A14; end; then g0 is continuous by Th20; hence thesis by A4; end; theorem Th32: for X being non empty TopSpace, f1 being map of X,R^1 st f1 is continuous ex g being map of X,R^1 st (for p being Point of X,r1 being real number st f1.p=r1 holds g.p=r1*r1) & g is continuous proof let X being non empty TopSpace,f1 be map of X,R^1; assume A1: f1 is continuous; defpred P[set,set] means (for r1 being real number st f1.$1=r1 holds $2=r1*r1); A2:for x being Element of X ex y being Element of REAL st P[x,y] proof let x be Element of X; reconsider r1=f1.x as Real by TOPMETR:24; set r3=r1*r1; for r1 being real number st f1.x=r1 holds r3=r1*r1; hence ex y being Element of REAL st (for r1 being real number st f1.x=r1 holds y=r1*r1); end; ex f being Function of the carrier of X,REAL st for x being Element of X holds P[x,f.x] from FuncExD(A2); then consider f being Function of the carrier of X,REAL such that A3: for x being Element of X holds (for r1 being real number st f1.x=r1 holds f.x=r1*r1); reconsider g0=f as map of X,R^1 by TOPMETR:24; A4: for p being Point of X,r1 being real number st f1.p=r1 holds g0.p=r1*r1 by A3; for p being Point of X,V being Subset of R^1 st g0.p in V & V is open holds ex W being Subset of X st p in W & W is open & g0.:W c= V proof let p be Point of X,V be Subset of R^1; assume A5:g0.p in V & V is open; reconsider r=g0.p as Real by TOPMETR:24; consider r0 being Real such that A6: r0>0 & ].r-r0,r+r0.[ c= V by A5,FRECHET:8; reconsider r1=f1.p as Real by TOPMETR:24; A7: r=r1*r1 by A3; then A8:r=r1^2 by SQUARE_1:def 3; then A9:0<=r by SQUARE_1:72; A10: r+r0>=r+0 by A6,REAL_1:55; then A11:(sqrt(r+r0))^2=r+r0 by A9,SQUARE_1:def 4; now per cases; case A12:r1>=0; then A13: r1=sqrt r by A8,A9,SQUARE_1:def 4; set r4=sqrt(r+r0)-sqrt(r); r+r0>r by A6,REAL_1:69; then sqrt(r+r0)>sqrt(r) by A9,SQUARE_1:95; then A14:r4>0 by SQUARE_1:11; r4^2=(sqrt(r+r0))^2-2*sqrt(r+r0)*sqrt(r)+(sqrt(r))^2 by SQUARE_1:64 .=r+r0-2*sqrt(r+r0)*sqrt(r)+(sqrt(r))^2 by A9,A10,SQUARE_1:def 4 .=r+r0-2*sqrt(r+r0)*sqrt(r)+r by A9,SQUARE_1:def 4 .=r+(r0-2*sqrt(r+r0)*sqrt(r))+r by XCMPLX_1:29 .=r+r+(r0-2*sqrt(r+r0)*sqrt(r)) by XCMPLX_1:1 .=2*r+(r0-2*sqrt(r+r0)*sqrt(r)) by XCMPLX_1:11 .=2*r+r0-2*sqrt(r+r0)*sqrt(r) by XCMPLX_1:29; then A15:2*r1*r4+r4^2= 2*r1*sqrt(r+r0)-2*r1*sqrt(r) +(2*r+r0-2*sqrt(r+r0)*sqrt(r)) by XCMPLX_1:40 .= (2*r1*sqrt(r+r0)-2*r1*r1) +(2*(r1*r1)+r0)-2*sqrt(r+r0)*r1 by A7,A13,XCMPLX_1:29 .= 2*r1*sqrt(r+r0)-2*r1*r1 +(2*(r1*r1)+r0)-2*(sqrt(r+r0)*r1) by XCMPLX_1:4 .= 2*r1*sqrt(r+r0)-2*r1*r1 +(2*r1*r1+r0)-2*(sqrt(r+r0)*r1) by XCMPLX_1:4 .= (2*r1*sqrt(r+r0)-2*r1*r1) +2*r1*r1+r0-2*(sqrt(r+r0)*r1) by XCMPLX_1:1 .= 2*r1*sqrt(r+r0)-(2*r1*r1 -2*r1*r1)+r0-2*(sqrt(r+r0)*r1) by XCMPLX_1:37 .= 2*r1*sqrt(r+r0)-0+r0-2*(sqrt(r+r0)*r1) by XCMPLX_1:14 .= 2*r1*sqrt(r+r0)+r0-2*r1*sqrt(r+r0) by XCMPLX_1:4 .=r0 by XCMPLX_1:26; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A16:r1<r1+r4 by A14,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A17:f1.p in G1 by A16,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A18: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A17,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A19: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A19; pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A20:f1.pz in f1.:W1 by A19,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A21:x=aa1*aa1 by A3,A19; then reconsider rx=x as Real; A22:r1-r4<aa1 & aa1<r1+r4 by A18,A20,JORDAN6:45; -0<=r1 by A12; then -r1<=0 by REAL_2:110; then -r1-r4<=r1-r4 by A12,REAL_1:49; then -(r4+r1)<=r1-r4 by XCMPLX_1:161; then -(r1+r4)<aa1 by A22,AXIOMS:22; then aa1--(r1+r4)>0 by SQUARE_1:11; then A23:r1+r4+aa1>0 by XCMPLX_1:151; r1+r4-aa1>0 by A22,SQUARE_1:11; then (r1+r4-aa1)*(r1+r4+aa1)>0 by A23,REAL_2:122; then (r1+r4)^2-aa1^2>0 by SQUARE_1:67; then A24: aa1^2<(r1+r4)^2 by REAL_2:106; (r1+r4)^2 =r1^2+2*r1*r4+r4^2 by SQUARE_1:63 .=r+2*r1*r4+r4^2 by A7,SQUARE_1:def 3 .=r+r0 by A15,XCMPLX_1:1; then A25:rx<r+r0 by A21,A24,SQUARE_1:def 3; aa1^2>=0 by SQUARE_1:72; then A26:0<=aa1*aa1 by SQUARE_1:def 3; now per cases; case 0<=r1-r4; then A27: (r1-r4)^2<aa1^2 by A22,SQUARE_1:78; r4^2>=0 by SQUARE_1:72; then (-2)*r4^2<=0 by REAL_2:123; then -2*r4^2<=0 by XCMPLX_1:175; then (r1-r4)^2 -aa1^2+-2*r4^2<= (r1-r4)^2 -aa1^2+0 by REAL_1:55; then (r1-r4)^2 +-2*r4^2 -aa1^2<= (r1-r4)^2 -aa1^2 by XCMPLX_1:29; then (r1-r4)^2 -2*r4^2 -aa1^2<= (r1-r4)^2 -aa1^2 by XCMPLX_0:def 8; then (r1-r4)^2 -2*r4^2 -aa1^2<0 by A27,REAL_2:105; then A28: aa1^2>(r1-r4)^2 -2*r4^2 by SQUARE_1:12; (r1-r4)^2 -2*r4^2=r1^2-2*r1*r4+r4^2-2*r4^2 by SQUARE_1:64 .=r1^2-2*r1*r4+r4^2-(r4^2+r4^2) by XCMPLX_1:11 .=r1^2-2*r1*r4+(r4^2-(r4^2+r4^2)) by XCMPLX_1:29 .=r1^2-2*r1*r4+(r4^2-r4^2-r4^2) by XCMPLX_1:36 .=r1^2-2*r1*r4+(0-r4^2) by XCMPLX_1:14 .=r-2*r1*r4+-r4^2 by A8,XCMPLX_1:150 .=r-2*r1*r4-r4^2 by XCMPLX_0:def 8 .=r-r0 by A15,XCMPLX_1:36; hence r-r0< aa1*aa1 by A28,SQUARE_1:def 3; case 0>r1-r4; then r1<r4 by SQUARE_1:12; then r1^2<r4^2 by A12,SQUARE_1:78; then A29:r1^2-r4^2<0 by REAL_2:105; 2*r1>=0 by A12,REAL_2:121; then 2*r1*r4>=0 by A14,REAL_2:121; then r1^2-r4^2 -2*r1*r4<0-0 by A29,REAL_1:92; hence r-r0< aa1*aa1 by A8,A15,A26,XCMPLX_1:36; end; hence x in ].r-r0,r+r0.[ by A21,A25,JORDAN6:45; end; then g0.:W c= V by A6,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A18; case A30:r1<0; then A31:-r1>0 by REAL_1:66; A32: (-r1)^2=r1^2 by SQUARE_1:61; then A33: -r1=sqrt r by A8,A31,SQUARE_1:89; A34:(sqrt(r))^2 =r1^2 by A8,A31,A32,SQUARE_1:89; set r4=sqrt(r+r0)-sqrt(r); r+r0>r by A6,REAL_1:69; then sqrt(r+r0)>sqrt(r) by A9,SQUARE_1:95; then A35:r4>0 by SQUARE_1:11; r4^2=r+r0-2*sqrt(r+r0)*sqrt(r)+(sqrt(r))^2 by A11,SQUARE_1:64 .=r+(r0-2*sqrt(r+r0)*sqrt(r))+r by A8,A34,XCMPLX_1:29 .=r+r+(r0-2*sqrt(r+r0)*sqrt(r)) by XCMPLX_1:1 .=2*r+(r0-2*sqrt(r+r0)*sqrt(r)) by XCMPLX_1:11 .=2*r+r0-2*sqrt(r+r0)*sqrt(r) by XCMPLX_1:29; then A36:-2*r1*r4+r4^2= -(2*r1*sqrt(r+r0)-2*r1*sqrt(r)) +(2*r+r0-2*sqrt(r+r0)*sqrt(r)) by XCMPLX_1:40 .= -(2*r1*sqrt(r+r0)-2*r1*(-r1)) +(2*(r1*r1)+r0)-2*sqrt(r+r0)*(-r1) by A7,A33,XCMPLX_1:29 .= -(2*r1*sqrt(r+r0)-2*(r1*(-r1))) +(2*(r1*r1)+r0)-2*sqrt(r+r0)*(-r1) by XCMPLX_1:4 .= -(2*r1*sqrt(r+r0)-2*(-(r1*r1))) +(2*(r1*r1)+r0)-2*sqrt(r+r0)*(-r1) by XCMPLX_1:175 .= -(2*r1*sqrt(r+r0)--2*(r1*r1)) +(2*(r1*r1)+r0)-2*sqrt(r+r0)*(-r1) by XCMPLX_1:175 .= -(2*r1*sqrt(r+r0)--(2*r1*r1)) +(2*(r1*r1)+r0)-2*sqrt(r+r0)*(-r1) by XCMPLX_1:4 .= -(2*r1*sqrt(r+r0)--(2*r1*r1)) +(2*(r1*r1)+r0)-2*(sqrt(r+r0)*(-r1)) by XCMPLX_1:4 .= -(2*r1*sqrt(r+r0)+2*r1*r1) +(2*(r1*r1)+r0)-2*(sqrt(r+r0)*(-r1)) by XCMPLX_1:151 .= -(2*r1*sqrt(r+r0)+2*r1*r1) +(2*(r1*r1)+r0)-2*(-(sqrt(r+r0)*r1)) by XCMPLX_1:175 .= -(2*r1*sqrt(r+r0)+2*r1*r1) +(2*(r1*r1)+r0)--(2*(sqrt(r+r0)*r1)) by XCMPLX_1:175 .= -(2*r1*sqrt(r+r0)+2*r1*r1) +(2*(r1*r1)+r0)+2*(sqrt(r+r0)*r1) by XCMPLX_1:151 .= -(2*r1*sqrt(r+r0)+2*r1*r1) +(2*r1*r1+r0)+2*(sqrt(r+r0)*r1) by XCMPLX_1:4 .= -(2*r1*sqrt(r+r0)+2*r1*r1) +2*r1*r1+r0+2*(sqrt(r+r0)*r1) by XCMPLX_1:1 .= (-2*r1*sqrt(r+r0)-2*r1*r1) +2*r1*r1+r0+2*(sqrt(r+r0)*r1) by XCMPLX_1:161 .= (-2*r1*sqrt(r+r0)+2*r1*r1) -2*r1*r1+r0+2*(sqrt(r+r0)*r1) by XCMPLX_1:29 .= -2*r1*sqrt(r+r0)+(2*r1*r1 -2*r1*r1)+r0+2*(sqrt(r+r0)*r1) by XCMPLX_1:29 .= -2*r1*sqrt(r+r0)+0+r0+2*(sqrt(r+r0)*r1) by XCMPLX_1:14 .= -2*r1*sqrt(r+r0)+r0+2*r1*sqrt(r+r0) by XCMPLX_1:4 .= 2*r1*sqrt(r+r0)+ -2*r1*sqrt(r+r0)+r0 by XCMPLX_1:1 .=r0 by XCMPLX_1:138; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A37:r1<r1+r4 by A35,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A38:f1.p in G1 by A37,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A39: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A38,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A40: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A40; pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A41:f1.pz in f1.:W1 by A40,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A42:x=aa1*aa1 by A3,A40; then reconsider rx=x as Real; A43:r1-r4<aa1 & aa1<r1+r4 by A39,A41,JORDAN6:45; -r1>=r1 by A30,A31,AXIOMS:22; then -r1-r4>=r1-r4 by REAL_1:49; then -(-r1-r4)<=-(r1-r4) by REAL_1:50; then r1+r4<=-(r1-r4) by XCMPLX_1:164; then -(r1-r4)>aa1 by A43,AXIOMS:22; then -(r1-r4)+(r1-r4)>aa1+(r1-r4) by REAL_1:67; then (r1-r4)-(r1-r4)>aa1+(r1-r4) by XCMPLX_0:def 8; then A44:r1-r4+aa1<0 by XCMPLX_1:14; aa1-(r1-r4)>0 by A43,SQUARE_1:11; then -(-aa1+(r1-r4))>0 by XCMPLX_1:163; then ((r1-r4)+-aa1)<0 by REAL_1:66; then r1-r4-aa1<0 by XCMPLX_0:def 8; then (r1-r4-aa1)*(r1-r4+aa1)>0 by A44,REAL_2:122; then (r1-r4)^2-aa1^2>0 by SQUARE_1:67; then A45: aa1^2<(r1-r4)^2 by REAL_2:106; (r1-r4)^2 =r1^2-2*r1*r4+r4^2 by SQUARE_1:64 .=r+-2*r1*r4+r4^2 by A8,XCMPLX_0:def 8 .=r+r0 by A36,XCMPLX_1:1; then A46:rx<r+r0 by A42,A45,SQUARE_1:def 3; aa1^2>=0 by SQUARE_1:72; then A47:0<=aa1*aa1 by SQUARE_1:def 3; now per cases; case 0>=r1+r4; then A48:-0<=-(r1+r4) by REAL_1:50; -aa1>-(r1+r4) by A43,REAL_1:50; then (-(r1+r4))^2<(-aa1)^2 by A48,SQUARE_1:78; then (r1+r4)^2<(-aa1)^2 by SQUARE_1:61; then A49: (r1+r4)^2<aa1^2 by SQUARE_1:61; r4^2>=0 by SQUARE_1:72; then (-2)*r4^2<=0 by REAL_2:123; then -2*r4^2<=0 by XCMPLX_1:175; then (r1+r4)^2 -aa1^2+-2*r4^2<= (r1+r4)^2 -aa1^2+0 by REAL_1:55; then (r1+r4)^2 +-2*r4^2 -aa1^2<= (r1+r4)^2 -aa1^2 by XCMPLX_1:29; then (r1+r4)^2 -2*r4^2 -aa1^2<= (r1+r4)^2 -aa1^2 by XCMPLX_0:def 8; then (r1+r4)^2 -2*r4^2 -aa1^2<0 by A49,REAL_2:105; then A50: aa1^2>(r1+r4)^2 -2*r4^2 by SQUARE_1:12; (r1+r4)^2 -2*r4^2=r1^2+2*r1*r4+r4^2-2*r4^2 by SQUARE_1:63 .=r1^2+2*r1*r4+r4^2-(r4^2+r4^2) by XCMPLX_1:11 .=r1^2+2*r1*r4+(r4^2-(r4^2+r4^2)) by XCMPLX_1:29 .=r1^2+2*r1*r4+(r4^2-r4^2-r4^2) by XCMPLX_1:36 .=r1^2+2*r1*r4+(0-r4^2) by XCMPLX_1:14 .=r+2*r1*r4+-r4^2 by A8,XCMPLX_1:150 .=r+2*r1*r4-r4^2 by XCMPLX_0:def 8 .=r--2*r1*r4-r4^2 by XCMPLX_1:151 .=r-r0 by A36,XCMPLX_1:36; hence r-r0< aa1*aa1 by A50,SQUARE_1:def 3; case 0<r1+r4; then 0+-r1<(r1+r4)+-r1 by REAL_1:67; then -r1<r4 by XCMPLX_1:137; then (-r1)^2<r4^2 by A31,SQUARE_1:78; then r1^2<r4^2 by SQUARE_1:61; then r1^2 -r1^2>r1^2-r4^2 by REAL_1:92; then A51:r1^2-r4^2<0 by XCMPLX_1:14; 2*r1<=0 by A30,REAL_2:123; then 2*r1*r4<=0 by A35,REAL_2:123; then A52:r1^2-r4^2 +2*r1*r4<0+0 by A51,REAL_1:67; r1^2-r4^2 +2*r1*r4=r+2*r1*r4-r4^2 by A8,XCMPLX_1:29 .=r--2*r1*r4-r4^2 by XCMPLX_1:151 .=r-r0 by A36,XCMPLX_1:36; hence r-r0< aa1*aa1 by A47,A52; end; hence x in ].r-r0,r+r0.[ by A42,A46,JORDAN6:45; end; then g0.:W c= V by A6,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A39; end; hence ex W being Subset of X st p in W & W is open & g0.:W c= V; end; then g0 is continuous by Th20; hence thesis by A4; end; theorem Th33: for X being non empty TopSpace, f1 being map of X,R^1,a being real number st f1 is continuous holds ex g being map of X,R^1 st (for p being Point of X,r1 being real number st f1.p=r1 holds g.p=a*r1) & g is continuous proof let X being non empty TopSpace,f1 be map of X,R^1,a being real number; assume A1: f1 is continuous; defpred P[set,set] means (for r1 being Real st f1.$1=r1 holds $2=a*r1); A2:for x being Element of X ex y being Element of REAL st P[x,y] proof let x be Element of X; reconsider r1=f1.x as Real by TOPMETR:24; reconsider r3=a*r1 as Element of REAL by XREAL_0:def 1; for r1 being Real st f1.x=r1 holds r3=a*r1; hence ex y being Element of REAL st (for r1 being Real st f1.x=r1 holds y=a*r1); end; ex f being Function of the carrier of X,REAL st for x being Element of X holds P[x,f.x] from FuncExD(A2); then consider f being Function of the carrier of X,REAL such that A3: for x being Element of X holds (for r1 being Real st f1.x=r1 holds f.x=a*r1); reconsider g0=f as map of X,R^1 by TOPMETR:24; A4: for p being Point of X,r1 being real number st f1.p=r1 holds g0.p=a*r1 proof let p be Point of X, r1 be real number such that A5: f1.p=r1; reconsider r1 as Element of REAL by XREAL_0:def 1; g0.p=a*r1 by A3,A5; hence thesis; end; for p being Point of X,V being Subset of R^1 st g0.p in V & V is open holds ex W being Subset of X st p in W & W is open & g0.:W c= V proof let p be Point of X,V be Subset of R^1; assume A6:g0.p in V & V is open; reconsider r=g0.p as Real by TOPMETR:24; consider r0 being Real such that A7: r0>0 & ].r-r0,r+r0.[ c= V by A6,FRECHET:8; reconsider r1=f1.p as Real by TOPMETR:24; A8: r=a*r1 by A3; A9:r=a*r1 by A3; now per cases; case A10:a>=0; now per cases by A10; case A11:a>0; set r4=r0/a; A12:r4>0 by A7,A11,REAL_2:127; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A13:r1<r1+r4 by A12,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A14:f1.p in G1 by A13,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A15: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A14,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A16: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A16; pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A17:f1.pz in f1.:W1 by A16,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A18:x=a*aa1 by A3,A16; reconsider rx=x as Real by A16,XREAL_0:def 1; A19:r1-r4<aa1 & aa1<r1+r4 by A15,A17,JORDAN6:45; a*(r1+r4) =a*r1+a*r4 by XCMPLX_1:8 .=r+r0 by A8,A11,XCMPLX_1:88; then A20:rx<r+r0 by A11,A18,A19,REAL_1:70; A21:a*(r1-r4)<a*aa1 by A11,A19,REAL_1:70; a*(r1-r4) =a*r1-a*r4 by XCMPLX_1:40 .=r-r0 by A8,A11,XCMPLX_1:88; hence x in ].r-r0,r+r0.[ by A18,A20,A21,JORDAN6:45; end; then g0.:W c= V by A7,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A15; case A22:a=0; set r4=r0; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A23:r1<r1+r4 by A7,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A24:f1.p in G1 by A23,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A25: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A24,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A26: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A26; reconsider aa1=f1.pz as Real by TOPMETR:24; A27:x=a*aa1 by A3,A26 .=0 by A22; r-r0<0 & 0<r+r0 by A7,A9,A22,REAL_2:105; hence x in ].r-r0,r+r0.[ by A27,JORDAN6:45; end; then g0.:W c= V by A7,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A25; end; hence ex W being Subset of X st p in W & W is open & g0.:W c= V; case A28:a<0; then A29:-a>0 by REAL_1:66; A30:-a<>0 by A28,REAL_1:66; set r4=r0/(-a); A31:r4>0 by A7,A29,REAL_2:127; A32:(-a)*r4=r0 by A30,XCMPLX_1:88; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A33:r1<r1+r4 by A31,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A34:f1.p in G1 by A33,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A35: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A34,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A36: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A36; pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A37:f1.pz in f1.:W1 by A36,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A38:x=a*aa1 by A3,A36; A39:r1-r4<aa1 & aa1<r1+r4 by A35,A37,JORDAN6:45; then A40: (a)*aa1<(a)*(r1-r4) by A28,REAL_1:71; A41:(a)*(r1-r4) =(a)*r1-(a)*r4 by XCMPLX_1:40 .=a*r1+-(a*r4) by XCMPLX_0:def 8 .=a*r1+(-a)*r4 by XCMPLX_1:175 .=r+r0 by A4,A32; (a)*(r1+r4) =(a)*r1+(a)*r4 by XCMPLX_1:8 .=a*r1--(a*r4) by XCMPLX_1:151 .=a*r1-(-a)*r4 by XCMPLX_1:175 .=r-r0 by A4,A32; then r-r0< (a)*aa1 by A28,A39,REAL_1:71; hence x in ].r-r0,r+r0.[ by A38,A40,A41,JORDAN6:45; end; then g0.:W c= V by A7,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A35; end; hence ex W being Subset of X st p in W & W is open & g0.:W c= V; end; then g0 is continuous by Th20; hence thesis by A4; end; theorem Th34: for X being non empty TopSpace, f1 being map of X,R^1,a being real number st f1 is continuous holds ex g being map of X,R^1 st (for p being Point of X,r1 being real number st f1.p=r1 holds g.p=r1+a) & g is continuous proof let X being non empty TopSpace,f1 be map of X,R^1,a being real number; assume A1: f1 is continuous; defpred P[set,set] means (for r1 being Real st f1.$1=r1 holds $2=r1+a); A2:for x being Element of X ex y being Element of REAL st P[x,y] proof let x be Element of X; reconsider r1=f1.x as Real by TOPMETR:24; reconsider r2=a as Element of REAL by XREAL_0:def 1; set r3 =r1+r2; for r1 being Real st f1.x=r1 holds r3=r1+r2; hence ex y being Element of REAL st (for r1 being Real st f1.x=r1 holds y=r1+a); end; ex f being Function of the carrier of X,REAL st for x being Element of X holds P[x,f.x] from FuncExD(A2); then consider f being Function of the carrier of X,REAL such that A3: for x being Element of X holds (for r1 being Real st f1.x=r1 holds f.x=r1+a); reconsider g0=f as map of X,R^1 by TOPMETR:24; A4: for p being Point of X,r1 being real number st f1.p=r1 holds g0.p=r1+a proof let p be Point of X, r1 be real number such that A5: f1.p=r1; reconsider r1 as Element of REAL by XREAL_0:def 1; g0.p=r1+a by A3,A5; hence thesis; end; for p being Point of X,V being Subset of R^1 st g0.p in V & V is open holds ex W being Subset of X st p in W & W is open & g0.:W c= V proof let p be Point of X,V be Subset of R^1; assume A6:g0.p in V & V is open; reconsider r=g0.p as Real by TOPMETR:24; consider r0 being Real such that A7: r0>0 & ].r-r0,r+r0.[ c= V by A6,FRECHET:8; reconsider r1=f1.p as Real by TOPMETR:24; set r4=r0; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A8:r1<r1+r4 by A7,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A9:f1.p in G1 by A8,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A10: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A9,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A11: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A11; pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A12:f1.pz in f1.:W1 by A11,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A13:x=aa1+a by A3,A11; A14:r1-r4<aa1 & aa1<r1+r4 by A10,A12,JORDAN6:45; then A15: (r1+r4)+a>aa1+a by REAL_1:67; A16: (r1+r4)+a =a+r1+r4 by XCMPLX_1:1 .=r+r0 by A3; A17:(r1-r4)+a<aa1+a by A14,REAL_1:67; (r1-r4)+a =r1+a-r4 by XCMPLX_1:29 .=r-r0 by A3; hence x in ].r-r0,r+r0.[ by A13,A15,A16,A17,JORDAN6:45; end; then g0.:W c= V by A7,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A10; end; then g0 is continuous by Th20; hence thesis by A4; end; theorem Th35: for X being non empty TopSpace, f1,f2 being map of X,R^1 st f1 is continuous & f2 is continuous holds ex g being map of X,R^1 st (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g.p=r1*r2) & g is continuous proof let X be non empty TopSpace, f1,f2 be map of X,R^1; assume A1:f1 is continuous & f2 is continuous; then consider g1 being map of X,R^1 such that A2: (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g1.p=r1+r2) & g1 is continuous by Th29; consider g2 being map of X,R^1 such that A3: (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g2.p=r1-r2) & g2 is continuous by A1,Th31; consider g3 being map of X,R^1 such that A4: (for p being Point of X,r1 being real number st g1.p=r1 holds g3.p=r1*r1) & g3 is continuous by A2,Th32; consider g4 being map of X,R^1 such that A5: (for p being Point of X,r1 being real number st g2.p=r1 holds g4.p=r1*r1) & g4 is continuous by A3,Th32; consider g5 being map of X,R^1 such that A6: (for p being Point of X,r1,r2 being real number st g3.p=r1 & g4.p=r2 holds g5.p=r1-r2) & g5 is continuous by A4,A5,Th31; consider g6 being map of X,R^1 such that A7: (for p being Point of X,r1 being real number st g5.p=r1 holds g6.p=(1/4)*r1) & g6 is continuous by A6,Th33; for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g6.p=r1*r2 proof let p be Point of X,r1,r2 be real number; assume A8:f1.p=r1 & f2.p=r2; then A9:g1.p=r1+r2 by A2; A10:g2.p=r1-r2 by A3,A8; A11:g3.p=(r1+r2)*(r1+r2) by A4,A9 .=(r1+r2)^2 by SQUARE_1:def 3; g4.p=(r1-r2)*(r1-r2) by A5,A10.=(r1-r2)^2 by SQUARE_1:def 3; then g5.p= (r1+r2)^2 -(r1-r2)^2 by A6,A11; then g6.p=(1/4)*( (r1+r2)^2 -(r1-r2)^2) by A7 .=(1/4)*( r1^2+2*r1*r2+r2^2 -(r1-r2)^2) by SQUARE_1:63 .=(1/4)*( r1^2+2*r1*r2+r2^2 -(r1^2-2*r1*r2+r2^2)) by SQUARE_1:64 .=(1/4)*( r1^2+2*r1*r2+r2^2-r2^2 -(r1^2-2*r1*r2)) by XCMPLX_1:36 .=(1/4)*( r1^2+2*r1*r2 -(r1^2-2*r1*r2)) by XCMPLX_1:26 .=(1/4)*( r1^2+2*r1*r2 -r1^2+2*r1*r2) by XCMPLX_1:37 .=(1/4)*(2*r1*r2+2*r1*r2) by XCMPLX_1:26 .=(1/4)*(2*(r1*r2)+2*r1*r2) by XCMPLX_1:4 .=(1/4)*(2*(r1*r2)+2*(r1*r2)) by XCMPLX_1:4 .=(1/4)*(2*(2*(r1*r2))) by XCMPLX_1:11 .=(1/4)*(2*2*(r1*r2)) by XCMPLX_1:4 .=(1/4)*4*(r1*r2) by XCMPLX_1:4 .= r1*r2; hence g6.p=r1*r2; end; hence thesis by A7; end; theorem Th36: for X being non empty TopSpace, f1 being map of X,R^1 st f1 is continuous & (for q being Point of X holds f1.q<>0) holds ex g being map of X,R^1 st (for p being Point of X,r1 being real number st f1.p=r1 holds g.p=1/r1) & g is continuous proof let X being non empty TopSpace,f1 be map of X,R^1; assume A1: f1 is continuous & (for q being Point of X holds f1.q<>0); defpred P[set,set] means (for r1 being Real st f1.$1=r1 holds $2=1/r1); A2:for x being Element of X ex y being Element of REAL st P[x,y] proof let x be Element of X; reconsider r1=f1.x as Real by TOPMETR:24; set r3=1/r1; for r1 being Real st f1.x=r1 holds r3=1/r1; hence ex y being Element of REAL st (for r1 being Real st f1.x=r1 holds y=1/r1); end; ex f being Function of the carrier of X,REAL st for x being Element of X holds P[x,f.x] from FuncExD(A2); then consider f being Function of the carrier of X,REAL such that A3: for x being Element of X holds (for r1 being Real st f1.x=r1 holds f.x=1/r1); reconsider g0=f as map of X,R^1 by TOPMETR:24; A4: for p being Point of X,r1 being real number st f1.p=r1 holds g0.p=1/r1 proof let p be Point of X,r1 be real number such that A5: f1.p=r1; reconsider r1 as Element of REAL by XREAL_0:def 1; g0.p=1/r1 by A3,A5; hence thesis; end; for p being Point of X,V being Subset of R^1 st g0.p in V & V is open holds ex W being Subset of X st p in W & W is open & g0.:W c= V proof let p be Point of X,V be Subset of R^1; assume A6:g0.p in V & V is open; reconsider r=g0.p as Real by TOPMETR:24; consider r0 being Real such that A7: r0>0 & ].r-r0,r+r0.[ c= V by A6,FRECHET:8; reconsider r1=f1.p as Real by TOPMETR:24; A8:r1<>0 by A1; A9: r=1/r1 by A3; then A10:r=r1" by XCMPLX_1:217; now per cases; case A11: r1>=0; then A12:0<r by A8,A10,REAL_1:72; A13: r+r0>=r+0 by A7,REAL_1:55; then A14: r+r0>0 by A8,A10,A11,REAL_1:72; A15:r+r0<r+r0+r0 by A7,REAL_1:69; then A16:0<r+r0+r0 by A12,A13,AXIOMS:22; r1*(1/r*r)=r1*1 by A12,XCMPLX_1:88; then r1*r*(1/r)=r1 by XCMPLX_1:4; then A17: 1 *(1/r)=r1 by A8,A9,XCMPLX_1:88; set r4=r0/r/(r+r0); A18:r0/r>0 by A7,A12,REAL_2:127; A19: r<r+r0 by A7,REAL_1:69; then A20:0<r+r0 by A12,AXIOMS:22; then A21:r4>0 by A18,REAL_2:127; A22:r1-r4=1/r-r0/(r+r0)/r by A17,XCMPLX_1:48 .=(1-r0/(r+r0))/r by XCMPLX_1:121 .=((r+r0)/(r+r0)-r0/(r+r0))/r by A12,A19,XCMPLX_1:60 .=((r+r0-r0)/(r+r0))/r by XCMPLX_1:121 .=r/(r+r0)/r by XCMPLX_1:26; r/(r+r0)>0 by A12,A20,REAL_2:127; then A23:r1-r4>0 by A12,A22,REAL_2:127; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A24:r1<r1+r4 by A21,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A25:f1.p in G1 by A24,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A26: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A25,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A27: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A27; pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A28:f1.pz in f1.:W1 by A27,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A29:x=1/aa1 by A3,A27; A30:r1-r4<aa1 & aa1<r1+r4 by A26,A28,JORDAN6:45; then 0<aa1 by A23,AXIOMS:22; then A31:1/(1/r+r4)<1/aa1 by A17,A30,REAL_2:200; A32:0<r0^2 by A7,SQUARE_1:74; then 0+r0^2<r0^2+r0^2 by REAL_1:67; then 0<r0^2+r0^2 by A32,AXIOMS:22; then 0<r0 *r0+r0^2 by SQUARE_1:def 3; then 0<r0 *r0+r0 *r0 by SQUARE_1:def 3; then r0 *r< r0 *r+(r0 *r0+r0 *r0) by REAL_1:69; then r0 *r-(r0 *r0+r0 *r0)< r0 *r by REAL_1:84; then (r0 *r-(r0 *r0+r0 *r0))+ r*r<r*r+r0 *r by REAL_1:67; then r*r+r0 *r-(r0 *r0+r0 *r0)<r*r+r0 *r by XCMPLX_1:29; then r*r+r0 *r-r0 *r0-r0 *r0<r*r+r0 *r by XCMPLX_1:36; then r*r+r0 *r-r0 *r0-r0 *r0<r*(r+r0) by XCMPLX_1:8; then (r*r+r0 *r+r0 *r-r0 *r-r0 *r0)-r0 *r0<r*(r+r0) by XCMPLX_1:26; then (r*r+r0 *r+r0 *r-(r0 *r+r0 *r0))-r0 *r0<r*(r+r0) by XCMPLX_1:36 ; then r*r+r0 *r+r0 *r-(r0 *r+r0 *r0+r0 *r0)<r*(r+r0) by XCMPLX_1:36; then r*r+r0 *r+r0 *r-(r+r0+r0)*r0<r*(r+r0) by XCMPLX_1:9; then (r+r0+r0)*r-(r+r0+r0)*r0<r*(r+r0) by XCMPLX_1:9; then (r+r0+r0)*(r-r0)<r*(r+r0) by XCMPLX_1:40; then (r-r0)*(r+r0+r0)/(r+r0+r0)<r*(r+r0)/(r+r0+r0) by A16,REAL_1:73; then r-r0<r*(r+r0)/(r+r0+r0) by A14,A15,XCMPLX_1:90; then r-r0<r/((r+r0+r0)/(r+r0)) by XCMPLX_1:78; then r-r0<r/((r+r0)/(r+r0)+r0/(r+r0)) by XCMPLX_1:63; then r-r0<r*1/(1+r0/(r+r0)) by A12,A19,XCMPLX_1:60; then r-r0<1/((1+r0/(r+r0))/r) by XCMPLX_1:78; then r-r0<1/(1/r+r0/(r+r0)/r) by XCMPLX_1:63; then r-r0<1/(1/r+r0/r/(r+r0)) by XCMPLX_1:48; then A33: r-r0<1/aa1 by A31,AXIOMS:22; A34: 1/aa1<1/(r1-r4) by A23,A30,REAL_2:151; 1/(r1-r4) =1/(r1-r0 *r"/(r+r0)) by XCMPLX_0:def 9 .=1/(r1-r0 *(1/r)/(r+r0)) by XCMPLX_1:217 .=1/(r1-r0/((r+r0)/r1)) by A17,XCMPLX_1:78 .=1/(r1*1-r1*(r0/(r+r0))) by XCMPLX_1:82 .=1/((1-(r0/(r+r0)))*r1) by XCMPLX_1:40 .=1/(((r+r0)/(r+r0)-(r0/(r+r0)))*r1) by A12,A13,XCMPLX_1:60 .=1/((r+r0-r0)/(r+r0)*r1) by XCMPLX_1:121 .=1/(r/(r+r0)*r1) by XCMPLX_1:26 .=1/(r/((r+r0)/r1)) by XCMPLX_1:82 .=1/(r*r1/(r+r0)) by XCMPLX_1:78 .=(r+r0)/(r*r1)*1 by XCMPLX_1:81 .=(r+r0)/1 by A8,A10,XCMPLX_0:def 7 .=r+r0; hence x in ].r-r0,r+r0.[ by A29,A33,A34,JORDAN6:45; end; then g0.:W c= V by A7,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A26; case r1<0; then A35: r1"<0 by REAL_2:149; then A36:0<-r by A10,REAL_1:66; A37: -r+r0>=-r+0 by A7,REAL_1:55; then A38: -r+r0>0 by A10,A35,REAL_1:66; A39: -r+r0<-r+r0+r0 by A7,REAL_1:69; then A40:0<-r+r0+r0 by A36,A37,AXIOMS:22; r1*((-r)*(1/(-r)))=r1*1 by A36,XCMPLX_1:88; then r1*(-r)*(1/(-r))=r1 by XCMPLX_1:4; then (-(r*r1))*(1/(-r))=r1 by XCMPLX_1:175; then (-1)*(1/(-r))=r1 by A8,A9,XCMPLX_1:88; then A41: -(1 *(1/(-r)))=r1 by XCMPLX_1:175; then A42: -r1=1/(-r); set r4=r0/(-r)/(-r+r0); A43:r0/(-r)>0 by A7,A36,REAL_2:127; A44: -r<-r+r0 by A7,REAL_1:69; then A45:0<-r+r0 by A36,AXIOMS:22; then A46:r4>0 by A43,REAL_2:127; A47:r1+r4=-(1/(-r))+r0/(-r+r0)/(-r) by A41,XCMPLX_1:48 .=(-1)/(-r)+r0/(-r+r0)/(-r) by XCMPLX_1:188 .=(-1+r0/(-r+r0))/(-r) by XCMPLX_1:63 .=(-((-r+r0)/(-r+r0))+r0/(-r+r0))/(-r) by A36,A44,XCMPLX_1:60 .=((-(-r+r0))/(-r+r0)+r0/(-r+r0))/(-r) by XCMPLX_1:188 .=((-(-r+r0)+r0)/(-r+r0))/(-r) by XCMPLX_1:63 .=((r-r0+r0)/(-r+r0))/(-r) by XCMPLX_1:163 .=((r+r0-r0)/(-r+r0))/(-r) by XCMPLX_1:29 .=r/(-r+r0)/(-r) by XCMPLX_1:26; (-r)/(-r+r0)>0 by A36,A45,REAL_2:127; then -(r/(-r+r0))>0 by XCMPLX_1:188; then (r/(-r+r0))<0 by REAL_1:66; then A48: (r1+r4)<0 by A36,A47,REAL_2:128; reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:24; A49:r1<r1+r4 by A46,REAL_1:69; then r1-r4<r1 by REAL_1:84; then A50:f1.p in G1 by A49,JORDAN6:45; G1 is open by JORDAN6:46; then consider W1 being Subset of X such that A51: p in W1 & W1 is open & f1.:W1 c= G1 by A1,A50,Th20; set W=W1; g0.:W c= ].r-r0,r+r0.[ proof let x be set;assume x in g0.:W; then consider z being set such that A52: z in dom g0 & z in W & g0.z=x by FUNCT_1:def 12; reconsider pz=z as Point of X by A52; pz in the carrier of X; then pz in dom f1 by FUNCT_2:def 1; then A53:f1.pz in f1.:W1 by A52,FUNCT_1:def 12; reconsider aa1=f1.pz as Real by TOPMETR:24; A54:x=1/aa1 by A3,A52; A55:r1-r4<aa1 & aa1<r1+r4 by A51,A53,JORDAN6:45; then 0>aa1 by A48,AXIOMS:22; then A56:1/(-(1/(-r))-r4)>1/aa1 by A41,A55,REAL_2:200; A57:0<r0^2 by A7,SQUARE_1:74; then 0+r0^2<r0^2+r0^2 by REAL_1:67; then 0<r0^2+r0^2 by A57,AXIOMS:22; then 0<r0 *r0+r0^2 by SQUARE_1:def 3; then 0<r0 *r0+r0 *r0 by SQUARE_1:def 3; then r0 *(-r)< r0 *(-r)+(r0 *r0+r0 *r0) by REAL_1:69; then r0 *(-r)-(r0 *r0+r0 *r0)< r0 *(-r) by REAL_1:84; then (r0 *(-r)-(r0 *r0+r0 *r0))+ (-r)*(-r)<r0 *(-r)+(-r)*(-r) by REAL_1:67; then (-r)*(-r)+r0 *(-r)-(r0 *r0+r0 *r0)<(-r)*(-r)+r0 *(-r) by XCMPLX_1:29; then (-r)*(-r)+r0 *(-r)-r0 *r0-r0 *r0<(-r)*(-r)+r0 *(-r) by XCMPLX_1: 36; then (-r)*(-r)+r0 *(-r)-r0 *r0-r0 *r0<(-r)*((-r)+r0) by XCMPLX_1:8; then ((-r)*(-r)+r0 *(-r)+r0 *(-r)-r0 *(-r)-r0 *r0)-r0 *r0<(-r)*((-r) +r0) by XCMPLX_1:26; then ((-r)*(-r)+r0 *(-r)+r0 *(-r)-(r0 *(-r)+r0 *r0))-r0 *r0<(-r)*((- r)+r0) by XCMPLX_1:36; then (-r)*(-r)+r0 *(-r)+r0 *(-r)-(r0 *(-r)+r0 *r0+r0 *r0)<(-r)*((-r) +r0) by XCMPLX_1:36; then (-r)*(-r)+r0 *(-r)+r0 *(-r)-((-r)+r0+r0)*r0<(-r)*((-r)+r0) by XCMPLX_1:9; then ((-r)+r0+r0)*(-r)-((-r)+r0+r0)*r0<(-r)*((-r)+r0) by XCMPLX_1:9; then ((-r)+r0+r0)*((-r)-r0)<(-r)*((-r)+r0) by XCMPLX_1:40; then ((-r)-r0)*((-r)+r0+r0)/((-r)+r0+r0)<(-r)*((-r)+r0)/((-r)+r0+r0) by A40,REAL_1:73; then (-r)-r0<(-r)*((-r)+r0)/((-r)+r0+r0) by A38,A39,XCMPLX_1:90; then (-r)-r0<(-r)/(((-r)+r0+r0)/((-r)+r0)) by XCMPLX_1:78; then (-r)-r0<(-r)/(((-r)+r0)/((-r)+r0)+r0/((-r)+r0)) by XCMPLX_1:63; then (-r)-r0<(-r)*1/(1+r0/((-r)+r0)) by A36,A44,XCMPLX_1:60; then (-r)-r0<1/((1+r0/((-r)+r0))/(-r)) by XCMPLX_1:78; then (-r)-r0<1/(1/(-r)+r0/((-r)+r0)/(-r)) by XCMPLX_1:63; then (-r)-r0<1/(1/(-r)+r0/(-r)/((-r)+r0)) by XCMPLX_1:48; then -(r+r0)<1/(1/(-r)+r4) by XCMPLX_1:161; then (r+r0)>-(1/(1/(-r)+r4)) by REAL_2:109; then (r+r0)>(1/-(1/(-r)+r4)) by XCMPLX_1:189; then r+r0>1/(-(1/(-r))-r4) by XCMPLX_1:161; then A58: r+r0>1/aa1 by A56,AXIOMS:22; A59: 1/aa1>1/(r1+r4) by A48,A55,REAL_2:151; 1/(r1+r4) =1/(r1+r0 *(-r)"/(-r+r0)) by XCMPLX_0:def 9 .=1/(r1+r0 *(1/(-r))/(-r+r0)) by XCMPLX_1:217 .=1/(r1+(-(r1*r0))/(-r+r0)) by A42,XCMPLX_1:175 .=1/(r1+-((r1*r0)/(-r+r0))) by XCMPLX_1:188 .=1/(r1-((r1*r0))/(-r+r0)) by XCMPLX_0:def 8 .=1/(r1-r0/((-r+r0)/r1)) by XCMPLX_1:78 .=1/(r1*1-r1*(r0/(-r+r0))) by XCMPLX_1:82 .=1/(r1*(1-r0/(-r+r0))) by XCMPLX_1:40 .=1/(((-r+r0)/(-r+r0)-(r0/(-r+r0)))*r1) by A36,A37,XCMPLX_1: 60 .=1/((-r+r0-r0)/(-r+r0)*(r1)) by XCMPLX_1:121 .=1/((-r+r0-r0)/(-(r-r0))*(r1)) by XCMPLX_1:162 .=1/((-(-r+r0-r0)/(r-r0))*(r1)) by XCMPLX_1:189 .=1/((-r+r0-r0)/((r-r0))*(-r1)) by XCMPLX_1:176 .=1/((-r)/((r-r0))*(-r1)) by XCMPLX_1:26 .=1/((-r)/((r-r0)/(-r1))) by XCMPLX_1:82 .=1/((-r)*(-r1)/(r-r0)) by XCMPLX_1:78 .=(r-r0)/((-r)*(-r1))*1 by XCMPLX_1:81 .=(r-r0)/((-r)*(-r)") by A41,XCMPLX_1:217 .=(r-r0)/1 by A36,XCMPLX_0:def 7 .=r-r0; hence x in ].r-r0,r+r0.[ by A54,A58,A59,JORDAN6:45; end; then g0.:W c= V by A7,XBOOLE_1:1; hence ex W being Subset of X st p in W & W is open & g0.:W c= V by A51; end; hence ex W being Subset of X st p in W & W is open & g0.:W c= V; end; then g0 is continuous by Th20; hence thesis by A4; end; theorem Th37: for X being non empty TopSpace, f1,f2 being map of X,R^1 st f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0) holds ex g being map of X,R^1 st (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g.p=r1/r2) & g is continuous proof let X be non empty TopSpace, f1,f2 be map of X,R^1; assume A1:f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0); then consider g1 being map of X,R^1 such that A2: (for p being Point of X,r2 being real number st f2.p=r2 holds g1.p=1/r2) & g1 is continuous by Th36; consider g2 being map of X,R^1 such that A3: (for p being Point of X,r1,r2 being real number st f1.p=r1 & g1.p=r2 holds g2.p=r1*r2) & g2 is continuous by A1,A2,Th35; for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g2.p=r1/r2 proof let p be Point of X,r1,r2 be real number; assume A4:f1.p=r1 & f2.p=r2; then g1.p=1/r2 by A2; then g2.p=r1*(1/r2) by A3,A4 .=r1/r2 by XCMPLX_1:100; hence g2.p=r1/r2; end; hence thesis by A3; end; theorem Th38: for X being non empty TopSpace, f1,f2 being map of X,R^1 st f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0) holds ex g being map of X,R^1 st (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g.p=r1/r2/r2) & g is continuous proof let X be non empty TopSpace, f1,f2 be map of X,R^1; assume A1:f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0); then consider g2 being map of X,R^1 such that A2: (for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g2.p=r1/r2) & g2 is continuous by Th37; consider g3 being map of X,R^1 such that A3: (for p being Point of X,r1,r2 being real number st g2.p=r1 & f2.p=r2 holds g3.p=r1/r2) & g3 is continuous by A1,A2,Th37; for p being Point of X,r1,r2 being real number st f1.p=r1 & f2.p=r2 holds g3.p=r1/r2/r2 proof let p be Point of X,r1,r2 be real number; assume A4:f1.p=r1 & f2.p=r2; then g2.p=r1/r2 by A2; hence g3.p=r1/r2/r2 by A3,A4; end; hence thesis by A3; end; theorem Th39: for K0 being Subset of TOP-REAL 2, f being map of (TOP-REAL 2)|K0,R^1 st (for p being Point of (TOP-REAL 2)|K0 holds f.p=proj1.p) holds f is continuous proof let K0 be Subset of TOP-REAL 2, f be map of (TOP-REAL 2)|K0,R^1; assume A1: (for p being Point of (TOP-REAL 2)|K0 holds f.p=proj1.p); A2:dom f= the carrier of (TOP-REAL 2)|K0 by FUNCT_2:def 1; A3: the carrier of (TOP-REAL 2)|K0 =[#]((TOP-REAL 2)|K0) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; A4:(the carrier of TOP-REAL 2)/\K0=K0 by XBOOLE_1:28; A5:for x being set st x in dom f holds f.x=proj1.x by A1; reconsider g=proj1 as map of TOP-REAL 2,R^1 by TOPMETR:24; A6:f=g|K0 by A2,A3,A4,A5,Th14,FUNCT_1:68; g is continuous by TOPREAL6:83; hence f is continuous by A6,TOPMETR:10; end; theorem Th40: for K0 being Subset of TOP-REAL 2, f being map of (TOP-REAL 2)|K0,R^1 st (for p being Point of (TOP-REAL 2)|K0 holds f.p=proj2.p) holds f is continuous proof let K0 be Subset of TOP-REAL 2, f be map of (TOP-REAL 2)|K0,R^1; assume A1: (for p being Point of (TOP-REAL 2)|K0 holds f.p=proj2.p); A2:dom f= the carrier of (TOP-REAL 2)|K0 by FUNCT_2:def 1; A3: the carrier of (TOP-REAL 2)|K0 =[#]((TOP-REAL 2)|K0) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; A4:(the carrier of TOP-REAL 2) /\ K0=K0 by XBOOLE_1:28; for x being set st x in dom f holds f.x=proj2.x by A1; then A5:f=proj2|K0 by A2,A3,A4,Th15,FUNCT_1:68; reconsider g=proj2 as map of TOP-REAL 2,R^1 by TOPMETR:24; g is continuous by TOPREAL6:83; hence f is continuous by A5,TOPMETR:10; end; theorem Th41: for K1 being non empty Subset of TOP-REAL 2, f being map of (TOP-REAL 2)|K1,R^1 st (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=1/p`1) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 ) holds f is continuous proof let K1 be non empty Subset of TOP-REAL 2, f be map of (TOP-REAL 2)|K1,R^1; assume A1:(for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=1/p`1) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 ); A2: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; proj1|K1 is Function of K1,the carrier of R^1 by FUNCT_2:38,TOPMETR:24; then reconsider g1=proj1|K1 as map of (TOP-REAL 2)|K1,R^1 by A2; A3:for q being Point of (TOP-REAL 2)|K1 holds g1.q=proj1.q proof let q be Point of (TOP-REAL 2)|K1; A4:q in the carrier of (TOP-REAL 2)|K1; dom proj1=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then q in dom proj1 /\ K1 by A2,A4,XBOOLE_0:def 3; hence g1.q=proj1.q by FUNCT_1:71; end; then A5:g1 is continuous by Th39; for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 proof let q be Point of (TOP-REAL 2)|K1; q in the carrier of (TOP-REAL 2)|K1; then reconsider q2=q as Point of TOP-REAL 2 by A2; g1.q=proj1.q by A3 .=q2`1 by PSCOMP_1:def 28; hence g1.q<>0 by A1; end; then consider g3 being map of (TOP-REAL 2)|K1,R^1 such that A6: (for q being Point of (TOP-REAL 2)|K1,r2 being real number st g1.q=r2 holds g3.q=1/r2) & g3 is continuous by A5,Th36; dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A7:dom f=dom g3 by FUNCT_2:def 1; for x being set st x in dom f holds f.x=g3.x proof let x be set;assume A8:x in dom f; then x in the carrier of (TOP-REAL 2)|K1; then x in [#]((TOP-REAL 2)|K1) by PRE_TOPC:12; then x in K1 by PRE_TOPC:def 10; then reconsider r=x as Point of (TOP-REAL 2); reconsider s=x as Point of (TOP-REAL 2)|K1 by A8; A9:f.r=1/r`1 by A1,A8; A10:g1.s=proj1.s by A3; proj1.r=r`1 by PSCOMP_1:def 28; hence f.x=g3.x by A6,A9,A10; end; hence f is continuous by A6,A7,FUNCT_1:9; end; theorem Th42: for K1 being non empty Subset of TOP-REAL 2, f being map of (TOP-REAL 2)|K1,R^1 st (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=1/p`2) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ) holds f is continuous proof let K1 be non empty Subset of TOP-REAL 2, f be map of (TOP-REAL 2)|K1,R^1; assume A1:(for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=1/p`2) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ); A2: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; proj2|K1 is Function of K1,the carrier of R^1 by FUNCT_2:38,TOPMETR:24; then reconsider g1=proj2|K1 as map of (TOP-REAL 2)|K1,R^1 by A2; A3:for q being Point of (TOP-REAL 2)|K1 holds g1.q=proj2.q proof let q be Point of (TOP-REAL 2)|K1; A4:q in the carrier of (TOP-REAL 2)|K1; dom proj2=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then q in dom proj2 /\ K1 by A2,A4,XBOOLE_0:def 3; hence g1.q=proj2.q by FUNCT_1:71; end; then A5:g1 is continuous by Th40; for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 proof let q be Point of (TOP-REAL 2)|K1; q in the carrier of (TOP-REAL 2)|K1; then reconsider q2=q as Point of TOP-REAL 2 by A2; g1.q=proj2.q by A3 .=q2`2 by PSCOMP_1:def 29; hence g1.q<>0 by A1; end; then consider g3 being map of (TOP-REAL 2)|K1,R^1 such that A6: (for q being Point of (TOP-REAL 2)|K1,r2 being real number st g1.q=r2 holds g3.q=1/r2) & g3 is continuous by A5,Th36; dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A7:dom f=dom g3 by FUNCT_2:def 1; for x being set st x in dom f holds f.x=g3.x proof let x be set;assume A8:x in dom f; then x in the carrier of (TOP-REAL 2)|K1; then x in [#]((TOP-REAL 2)|K1) by PRE_TOPC:12; then x in K1 by PRE_TOPC:def 10; then reconsider r=x as Point of (TOP-REAL 2); reconsider s=x as Point of (TOP-REAL 2)|K1 by A8; A9:f.r=1/r`2 by A1,A8; A10:g1.s=proj2.s by A3; proj2.r=r`2 by PSCOMP_1:def 29; hence f.x=g3.x by A6,A9,A10; end; hence f is continuous by A6,A7,FUNCT_1:9; end; theorem Th43: for K1 being non empty Subset of TOP-REAL 2, f being map of (TOP-REAL 2)|K1,R^1 st (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=p`2/p`1/p`1) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 ) holds f is continuous proof let K1 be non empty Subset of TOP-REAL 2, f be map of (TOP-REAL 2)|K1,R^1; assume A1:(for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=p`2/p`1/p`1) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 ); A2: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; proj2|K1 is Function of K1,the carrier of R^1 by FUNCT_2:38,TOPMETR:24; then reconsider g2=proj2|K1 as map of (TOP-REAL 2)|K1,R^1 by A2; A3:for q being Point of (TOP-REAL 2)|K1 holds g2.q=proj2.q proof let q be Point of (TOP-REAL 2)|K1; A4:q in the carrier of (TOP-REAL 2)|K1; dom proj2=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then q in dom proj2 /\ K1 by A2,A4,XBOOLE_0:def 3; hence g2.q=proj2.q by FUNCT_1:71; end; then A5:g2 is continuous by Th40; proj1|K1 is Function of K1,the carrier of R^1 by FUNCT_2:38,TOPMETR:24; then reconsider g1=proj1|K1 as map of (TOP-REAL 2)|K1,R^1 by A2; A6:for q being Point of (TOP-REAL 2)|K1 holds g1.q=proj1.q proof let q be Point of (TOP-REAL 2)|K1; A7:q in the carrier of (TOP-REAL 2)|K1; dom proj1=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then q in dom proj1 /\ K1 by A2,A7,XBOOLE_0:def 3; hence g1.q=proj1.q by FUNCT_1:71; end; then A8:g1 is continuous by Th39; for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 proof let q be Point of (TOP-REAL 2)|K1; q in the carrier of (TOP-REAL 2)|K1; then reconsider q2=q as Point of TOP-REAL 2 by A2; g1.q=proj1.q by A6 .=q2`1 by PSCOMP_1:def 28; hence g1.q<>0 by A1; end; then consider g3 being map of (TOP-REAL 2)|K1,R^1 such that A9: (for q being Point of (TOP-REAL 2)|K1,r1,r2 being real number st g2.q=r1 & g1.q=r2 holds g3.q=r1/r2/r2) & g3 is continuous by A5,A8,Th38; dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A10:dom f=dom g3 by FUNCT_2:def 1; for x being set st x in dom f holds f.x=g3.x proof let x be set;assume A11:x in dom f; then x in the carrier of (TOP-REAL 2)|K1; then x in [#]((TOP-REAL 2)|K1) by PRE_TOPC:12; then x in K1 by PRE_TOPC:def 10; then reconsider r=x as Point of (TOP-REAL 2); reconsider s=x as Point of (TOP-REAL 2)|K1 by A11; A12:f.r=r`2/r`1/r`1 by A1,A11; A13:g2.s=proj2.s by A3; A14:g1.s=proj1.s by A6; A15:proj2.r=r`2 by PSCOMP_1:def 29; proj1.r=r`1 by PSCOMP_1:def 28; hence f.x=g3.x by A9,A12,A13,A14,A15; end; hence f is continuous by A9,A10,FUNCT_1:9; end; theorem Th44: for K1 being non empty Subset of TOP-REAL 2, f being map of (TOP-REAL 2)|K1,R^1 st (for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=p`1/p`2/p`2) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ) holds f is continuous proof let K1 be non empty Subset of TOP-REAL 2, f be map of (TOP-REAL 2)|K1,R^1; assume A1:(for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=p`1/p`2/p`2) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ); A2: the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; proj1|K1 is Function of K1,the carrier of R^1 by FUNCT_2:38,TOPMETR:24; then reconsider g2=proj1|K1 as map of (TOP-REAL 2)|K1,R^1 by A2; A3:for q being Point of (TOP-REAL 2)|K1 holds g2.q=proj1.q proof let q be Point of (TOP-REAL 2)|K1; A4:q in the carrier of (TOP-REAL 2)|K1; dom proj1=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then q in dom proj1 /\ K1 by A2,A4,XBOOLE_0:def 3; hence g2.q=proj1.q by FUNCT_1:71; end; then A5:g2 is continuous by Th39; proj2|K1 is Function of K1,the carrier of R^1 by FUNCT_2:38,TOPMETR:24; then reconsider g1=proj2|K1 as map of (TOP-REAL 2)|K1,R^1 by A2; A6:for q being Point of (TOP-REAL 2)|K1 holds g1.q=proj2.q proof let q be Point of (TOP-REAL 2)|K1; A7:q in the carrier of (TOP-REAL 2)|K1; dom proj2=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then q in dom proj2 /\ K1 by A2,A7,XBOOLE_0:def 3; hence g1.q=proj2.q by FUNCT_1:71; end; then A8:g1 is continuous by Th40; for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 proof let q be Point of (TOP-REAL 2)|K1; q in the carrier of (TOP-REAL 2)|K1; then reconsider q2=q as Point of TOP-REAL 2 by A2; g1.q=proj2.q by A6 .=q2`2 by PSCOMP_1:def 29; hence g1.q<>0 by A1; end; then consider g3 being map of (TOP-REAL 2)|K1,R^1 such that A9: (for q being Point of (TOP-REAL 2)|K1,r1,r2 being real number st g2.q=r1 & g1.q=r2 holds g3.q=r1/r2/r2) & g3 is continuous by A5,A8,Th38; dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1; then A10:dom f=dom g3 by FUNCT_2:def 1; for x being set st x in dom f holds f.x=g3.x proof let x be set;assume A11:x in dom f; then x in the carrier of (TOP-REAL 2)|K1; then x in [#]((TOP-REAL 2)|K1) by PRE_TOPC:12; then x in K1 by PRE_TOPC:def 10; then reconsider r=x as Point of (TOP-REAL 2); reconsider s=x as Point of (TOP-REAL 2)|K1 by A11; A12:f.r=r`1/r`2/r`2 by A1,A11; A13:g2.s=proj1.s by A3; A14:g1.s=proj2.s by A6; A15:proj1.r=r`1 by PSCOMP_1:def 28; proj2.r=r`2 by PSCOMP_1:def 29; hence f.x=g3.x by A9,A12,A13,A14,A15; end; hence f is continuous by A9,A10,FUNCT_1:9; end; theorem Th45: for K0,B0 being Subset of TOP-REAL 2, f being map of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0, f1,f2 being map of (TOP-REAL 2)|K0,R^1 st f1 is continuous & f2 is continuous & K0<>{} & B0<>{} & (for x,y,r,s being real number st |[x,y]| in K0 & r=f1.(|[x,y]|) & s=f2.(|[x,y]|) holds f.(|[x,y]|)=|[r,s]|) holds f is continuous proof let K0,B0 be Subset of TOP-REAL 2, f be map of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0, f1,f2 be map of (TOP-REAL 2)|K0,R^1; assume A1:f1 is continuous & f2 is continuous & K0<>{} & B0<>{} & (for x,y,r,s being real number st |[x,y]| in K0 & r=f1.(|[x,y]|) & s=f2.(|[x,y]|) holds f. |[x,y]|=|[r,s]|); then reconsider K1=K0 as non empty Subset of TOP-REAL 2; reconsider B1=B0 as non empty Subset of TOP-REAL 2 by A1; reconsider X=(TOP-REAL 2)|K1,Y=(TOP-REAL 2)|B1 as non empty TopSpace; reconsider f0=f as map of X,Y; for r being Point of X,V being Subset of Y st f0.r in V & V is open holds ex W being Subset of X st r in W & W is open & f0.:W c= V proof let r be Point of X,V be Subset of Y; assume A2: f0.r in V & V is open; then consider V2 being Subset of TOP-REAL 2 such that A3: V2 is open & V=V2 /\ [#]Y by TOPS_2:32; A4:V2 /\ [#]Y c= V2 by XBOOLE_1:17; then f0.r in V2 by A2,A3; then reconsider p=f0.r as Point of TOP-REAL 2; consider r2 being real number such that A5: r2>0 & {q where q is Point of TOP-REAL 2: p`1-r2<q`1 & q`1<p`1+r2 & p`2-r2<q`2 & q`2<p`2+r2} c= V2 by A2,A3,A4,Th21; A6:r in the carrier of X; then A7:r in dom f1 by FUNCT_2:def 1; A8:r in dom f2 by A6,FUNCT_2:def 1; A9:f1.r in rng f1 by A7,FUNCT_1:12; f2.r in rng f2 by A8,FUNCT_1:12; then reconsider r3=f1.r,r4=f2.r as Real by A9,TOPMETR:24; A10:the carrier of X=[#]X by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; then r in K0; then reconsider pr=r as Point of TOP-REAL 2; A11:r= |[pr`1,pr`2]| by EUCLID:57; then A12:f0. |[pr`1,pr`2]|=|[r3,r4]| by A1,A10; p`1 <p`1+r2 by A5,REAL_1:69; then p`1-r2< p`1 & p`1<p`1+r2 by REAL_1:84; then A13: p`1 in ].p`1-r2,p`1+r2.[ by JORDAN6:45; then A14: f1.r in ].p`1-r2,p`1+r2.[ by A11,A12,EUCLID:56; p`2 <p`2+r2 by A5,REAL_1:69; then p`2-r2< p`2 & p`2<p`2+r2 by REAL_1:84; then A15:p`2 in ].p`2-r2,p`2+r2.[ by JORDAN6:45; reconsider G1= ].p`1-r2,p`1+r2.[,G2= ].p`2-r2,p`2+r2.[ as Subset of R^1 by TOPMETR:24; A16:G1 is open & G2 is open by JORDAN6:46; A17:f1.r in G1 & f2.r in G2 by A11,A12,A13,A15,EUCLID:56; consider W1 being Subset of X such that A18: r in W1 & W1 is open & f1.:W1 c= G1 by A1,A14,A16,Th20; consider W2 being Subset of X such that A19: r in W2 & W2 is open & f2.:W2 c= G2 by A1,A16,A17,Th20; reconsider W5=W1 /\ W2 as Subset of X; A20:W5 is open by A18,A19,TOPS_1:38; A21:r in W5 by A18,A19,XBOOLE_0:def 3; W5 c= W1 by XBOOLE_1:17; then f1.:W5 c= f1.:W1 by RELAT_1:156; then A22:f1.:W5 c= G1 by A18,XBOOLE_1:1; W5 c= W2 by XBOOLE_1:17; then f2.:W5 c= f2.:W2 by RELAT_1:156; then A23:f2.:W5 c= G2 by A19,XBOOLE_1:1; f0.:W5 c= V proof let v be set;assume A24:v in f0.:W5; then reconsider q2=v as Point of Y; consider k being set such that A25: k in dom f0 & k in W5 & q2=f0.k by A24,FUNCT_1:def 12; q2 in the carrier of Y; then A26:q2 in [#]Y by PRE_TOPC:12; the carrier of X=[#]X by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; then k in K0 by A25; then reconsider r8=k as Point of TOP-REAL 2; A27:dom f0=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1 .=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; A28:k= |[r8`1,r8`2]| by EUCLID:57; A29: |[r8`1,r8`2]| in K0 by A25,A27,EUCLID:57; A30:dom f1=the carrier of (TOP-REAL 2)|K0 by FUNCT_2:def 1 .=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; A31:dom f2=the carrier of (TOP-REAL 2)|K0 by FUNCT_2:def 1 .=[#]((TOP-REAL 2)|K0) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; A32:f1.(|[r8`1,r8`2]|) in rng f1 by A29,A30,FUNCT_1:def 5; f2.(|[r8`1,r8`2]|) in rng f2 by A29,A31,FUNCT_1:def 5; then reconsider r7=f1.(|[r8`1,r8`2]|), s7=f2.(|[r8`1,r8`2]|) as Real by A32,TOPMETR:24; A33:v=|[r7,s7]| by A1,A25,A27,A28; A34:(|[r7,s7]|)`1 =r7 by EUCLID:56; A35:(|[r7,s7]|)`2 =s7 by EUCLID:56; A36: |[r8`1,r8`2]| in W5 by A25,EUCLID:57; then A37: f1.(|[r8`1,r8`2]|) in f1.:W5 by A29,A30,FUNCT_1:def 12; f2.(|[r8`1,r8`2]|) in f2.:W5 by A29,A31,A36,FUNCT_1:def 12; then p`1-r2< r7 & r7<p`1+r2 & p`2-r2< s7 & s7<p`2+r2 by A22,A23,A37,JORDAN6:45; then v in {q3 where q3 is Point of TOP-REAL 2: p`1-r2<q3`1 & q3`1<p`1+r2 & p`2-r2<q3`2 & q3`2<p`2+r2} by A33,A34,A35; hence v in V by A3,A5,A26,XBOOLE_0:def 3; end; hence ex W being Subset of X st r in W & W is open & f0.:W c= V by A20,A21; end; hence f is continuous by Th20; end; theorem Th46: for K0,B0 being Subset of TOP-REAL 2,f being map of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} holds f is continuous proof let K0,B0 be Subset of TOP-REAL 2,f be map of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0; assume A1:f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2}; ((1.REAL 2)`2<=(1.REAL 2)`1 & -(1.REAL 2)`1<=(1.REAL 2)`2 or (1.REAL 2)`2>=(1.REAL 2)`1 & (1.REAL 2)`2<=-(1.REAL 2)`1) & (1.REAL 2)<>0.REAL 2 by Th13,REVROT_1:19; then A2:1.REAL 2 in K0 by A1; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A3: K0 c= B0 proof let x be set;assume x in K0; then consider p8 being Point of TOP-REAL 2 such that A4: x=p8 & ( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2) by A1; not x in {0.REAL 2} by A4,TARSKI:def 1; hence x in B0 by A1,A4,XBOOLE_0:def 4; end; A5:dom ((proj2)*(Out_In_Sq|K1)) c= dom (Out_In_Sq|K1) by RELAT_1:44; A6:dom (Out_In_Sq|K1) c= dom ((proj2)*(Out_In_Sq|K1)) proof let x be set;assume A7:x in dom (Out_In_Sq|K1); then A8:x in dom Out_In_Sq /\ K1 by FUNCT_1:68; A9:(Out_In_Sq|K1).x=Out_In_Sq.x by A7,FUNCT_1:68; A10: dom proj2 = (the carrier of TOP-REAL 2) by FUNCT_2:def 1; x in dom Out_In_Sq by A8,XBOOLE_0:def 3; then Out_In_Sq.x in rng Out_In_Sq by FUNCT_1:12; then Out_In_Sq.x in the carrier of TOP-REAL 2 by XBOOLE_0:def 4; hence x in dom ((proj2)*(Out_In_Sq|K1)) by A7,A9,A10,FUNCT_1:21; end; A11:K1 c= (the carrier of TOP-REAL 2)\{0.REAL 2} proof let z be set;assume z in K1; then consider p8 being Point of TOP-REAL 2 such that A12: p8=z &( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2) by A1; z in (the carrier of TOP-REAL 2) & not z in {0.REAL 2} by A12,TARSKI:def 1; hence thesis by XBOOLE_0:def 4; end; A13: (the carrier of TOP-REAL 2)\{0.REAL 2}<>{} by Th19; A14:dom ((proj2)*(Out_In_Sq|K1)) =dom (Out_In_Sq|K1) by A5,A6,XBOOLE_0:def 10 .=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A13,FUNCT_2:def 1 .=K1 by A11,XBOOLE_1:28 .=[#]((TOP-REAL 2)|K1) by PRE_TOPC:def 10 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:12; rng ((proj2)*(Out_In_Sq|K1)) c= rng (proj2) by RELAT_1:45; then rng ((proj2)*(Out_In_Sq|K1)) c= the carrier of R^1 by TOPMETR:24,XBOOLE_1: 1; then (proj2)*(Out_In_Sq|K1) is Function of the carrier of (TOP-REAL 2)|K1, the carrier of R^1 by A14,FUNCT_2:4; then reconsider g2=(proj2)*(Out_In_Sq|K1) as map of (TOP-REAL 2)|K1,R^1; A15:dom ((proj1)*(Out_In_Sq|K1)) c= dom (Out_In_Sq|K1) by RELAT_1:44; dom (Out_In_Sq|K1) c= dom ((proj1)*(Out_In_Sq|K1)) proof let x be set;assume A16:x in dom (Out_In_Sq|K1); then A17:x in dom Out_In_Sq /\ K1 by FUNCT_1:68; A18:(Out_In_Sq|K1).x=Out_In_Sq.x by A16,FUNCT_1:68; A19: dom proj1 = (the carrier of TOP-REAL 2) by FUNCT_2:def 1; x in dom Out_In_Sq by A17,XBOOLE_0:def 3; then Out_In_Sq.x in rng Out_In_Sq by FUNCT_1:12; then Out_In_Sq.x in the carrier of TOP-REAL 2 by XBOOLE_0:def 4; hence x in dom ((proj1)*(Out_In_Sq|K1)) by A16,A18,A19,FUNCT_1:21; end; then A20:dom ((proj1)*(Out_In_Sq|K1)) =dom (Out_In_Sq|K1) by A15,XBOOLE_0:def 10 .=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A13,FUNCT_2:def 1 .=K1 by A11,XBOOLE_1:28 .=[#]((TOP-REAL 2)|K1) by PRE_TOPC:def 10 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:12; rng ((proj1)*(Out_In_Sq|K1)) c= rng (proj1) by RELAT_1:45; then rng ((proj1)*(Out_In_Sq|K1)) c= the carrier of R^1 by TOPMETR:24,XBOOLE_1: 1; then (proj1)*(Out_In_Sq|K1) is Function of the carrier of (TOP-REAL 2)|K1, the carrier of R^1 by A20,FUNCT_2:4; then reconsider g1=(proj1)*(Out_In_Sq|K1) as map of (TOP-REAL 2)|K1,R^1; A21: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 proof let q be Point of TOP-REAL 2; assume A22:q in the carrier of (TOP-REAL 2)|K1; the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A23: q=p3 & ((p3`2<=p3`1 & -p3`1<=p3`2 or p3`2>=p3`1 & p3`2<=-p3`1) & p3<>0.REAL 2) by A1,A22; now assume A24:q`1=0; then q`2=0 by A23; hence contradiction by A23,A24,EUCLID:57,58; end; hence q`1<>0; end; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=p`2/p`1/p`1 proof let p be Point of TOP-REAL 2; assume A25: p in the carrier of (TOP-REAL 2)|K1; A26: (the carrier of TOP-REAL 2)\{0.REAL 2}<>{} by Th19; A27:dom (Out_In_Sq|K1)=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A26,FUNCT_2:def 1 .=K1 by A11,XBOOLE_1:28; A28:the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A29: p=p3 & ((p3`2<=p3`1 & -p3`1<=p3`2 or p3`2>=p3`1 & p3`2<=-p3`1) & p3<>0.REAL 2) by A1,A25; A30:Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by A29,Def1; (Out_In_Sq|K1).p=Out_In_Sq.p by A25,A28,FUNCT_1:72; then g2.p=(proj2).(|[1/p`1,p`2/p`1/p`1]|) by A25,A27,A28,A30,FUNCT_1:23 .=(|[1/p`1,p`2/p`1/p`1]|)`2 by PSCOMP_1:def 29 .=p`2/p`1/p`1 by EUCLID:56; hence g2.p=p`2/p`1/p`1; end; then consider f2 being map of (TOP-REAL 2)|K1,R^1 such that A31:for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds f2.p=p`2/p`1/p`1; A32:f2 is continuous by A21,A31,Th43; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=1/p`1 proof let p be Point of TOP-REAL 2; assume A33: p in the carrier of (TOP-REAL 2)|K1; A34:K1 c= (the carrier of TOP-REAL 2)\{0.REAL 2} proof let z be set;assume z in K1; then consider p8 being Point of TOP-REAL 2 such that A35: p8=z &( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2) by A1; z in (the carrier of TOP-REAL 2) & not z in {0.REAL 2} by A35,TARSKI:def 1; hence z in (the carrier of TOP-REAL 2)\{0.REAL 2} by XBOOLE_0:def 4; end; A36: (the carrier of TOP-REAL 2)\{0.REAL 2}<>{} by Th19; A37:dom (Out_In_Sq|K1)=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A36,FUNCT_2:def 1 .=K1 by A34,XBOOLE_1:28; A38:the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A39: p=p3 & ((p3`2<=p3`1 & -p3`1<=p3`2 or p3`2>=p3`1 & p3`2<=-p3`1) & p3<>0.REAL 2) by A1,A33; A40:Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by A39,Def1; (Out_In_Sq|K1).p=Out_In_Sq.p by A33,A38,FUNCT_1:72; then g1.p=(proj1).(|[1/p`1,p`2/p`1/p`1]|) by A33,A37,A38,A40,FUNCT_1:23 .=(|[1/p`1,p`2/p`1/p`1]|)`1 by PSCOMP_1:def 28 .=1/p`1 by EUCLID:56; hence g1.p=1/p`1; end; then consider f1 being map of (TOP-REAL 2)|K1,R^1 such that A41:for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds f1.p=1/p`1; A42:f1 is continuous by A21,A41,Th41; for x,y,r,s being real number st |[x,y]| in K1 & r=f1.(|[x,y]|) & s=f2.(|[x,y]|) holds f. |[x,y]|=|[r,s]| proof let x,y,r,s be real number; assume A43: |[x,y]| in K1 & r=f1.(|[x,y]|) & s=f2.(|[x,y]|); set p99=|[x,y]|; A44:the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; consider p3 being Point of TOP-REAL 2 such that A45: p99=p3 & ((p3`2<=p3`1 & -p3`1<=p3`2 or p3`2>=p3`1 & p3`2<=-p3`1) & p3<>0.REAL 2) by A1,A43; A46:((p99`2<=p99`1 & -p99`1<=p99`2 or p99`2>=p99`1 & p99`2<=-p99`1) implies Out_In_Sq.p99=|[1/p99`1,p99`2/p99`1/p99`1]|) & (not(p99`2<=p99`1 & -p99`1<=p99`2 or p99`2>=p99`1 & p99`2<=-p99`1) implies Out_In_Sq.p99=|[p99`1/p99`2/p99`2,1/p99`2]|) by A45,Def1; A47:f1.p99=1/p99`1 by A41,A43,A44; (Out_In_Sq|K0). |[x,y]|= |[1/p99`1,p99`2/p99`1/p99`1]| by A43,A45,A46, FUNCT_1:72 .=|[r,s]| by A31,A43,A44,A47; hence f. |[x,y]|=|[r,s]| by A1; end; hence f is continuous by A2,A3,A32,A42,Th45; end; theorem Th47: for K0,B0 being Subset of TOP-REAL 2,f being map of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} holds f is continuous proof let K0,B0 be Subset of TOP-REAL 2,f be map of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0; assume A1:f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2}; ((1.REAL 2)`1<=(1.REAL 2)`2 & -(1.REAL 2)`2<=(1.REAL 2)`1 or (1.REAL 2)`1>=(1.REAL 2)`2 & (1.REAL 2)`1<=-(1.REAL 2)`2) & (1.REAL 2)<>0.REAL 2 by Th13,REVROT_1:19; then A2:1.REAL 2 in K0 by A1; then reconsider K1=K0 as non empty Subset of TOP-REAL 2; A3: K0 c= B0 proof let x be set;assume x in K0; then consider p8 being Point of TOP-REAL 2 such that A4: x=p8 & ( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2) by A1; not x in {0.REAL 2} by A4,TARSKI:def 1; hence x in B0 by A1,A4,XBOOLE_0:def 4; end; A5:dom ((proj1)*(Out_In_Sq|K1)) c= dom (Out_In_Sq|K1) by RELAT_1:44; A6:dom (Out_In_Sq|K1) c= dom ((proj1)*(Out_In_Sq|K1)) proof let x be set;assume A7:x in dom (Out_In_Sq|K1); then A8:x in dom Out_In_Sq /\ K1 by FUNCT_1:68; A9:(Out_In_Sq|K1).x=Out_In_Sq.x by A7,FUNCT_1:68; A10: dom proj1 = (the carrier of TOP-REAL 2) by FUNCT_2:def 1; x in dom Out_In_Sq by A8,XBOOLE_0:def 3; then Out_In_Sq.x in rng Out_In_Sq by FUNCT_1:12; then Out_In_Sq.x in the carrier of TOP-REAL 2 by XBOOLE_0:def 4; hence x in dom ((proj1)*(Out_In_Sq|K1)) by A7,A9,A10,FUNCT_1:21; end; A11:K1 c= (the carrier of TOP-REAL 2)\{0.REAL 2} proof let z be set;assume z in K1; then consider p8 being Point of TOP-REAL 2 such that A12: p8=z &( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2) by A1; z in (the carrier of TOP-REAL 2) & not z in {0.REAL 2} by A12,TARSKI:def 1; hence z in (the carrier of TOP-REAL 2)\{0.REAL 2} by XBOOLE_0:def 4; end; A13: (the carrier of TOP-REAL 2)\{0.REAL 2}<>{} by Th19; A14:dom ((proj1)*(Out_In_Sq|K1)) =dom (Out_In_Sq|K1) by A5,A6,XBOOLE_0:def 10 .=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A13,FUNCT_2:def 1 .=K1 by A11,XBOOLE_1:28 .=[#]((TOP-REAL 2)|K1) by PRE_TOPC:def 10 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:12; rng ((proj1)*(Out_In_Sq|K1)) c= rng (proj1) by RELAT_1:45; then rng ((proj1)*(Out_In_Sq|K1)) c= the carrier of R^1 by TOPMETR:24,XBOOLE_1: 1; then (proj1)*(Out_In_Sq|K1) is Function of the carrier of (TOP-REAL 2)|K1, the carrier of R^1 by A14,FUNCT_2:4; then reconsider g2=(proj1)*(Out_In_Sq|K1) as map of (TOP-REAL 2)|K1,R^1; A15:dom ((proj2)*(Out_In_Sq|K1)) c= dom (Out_In_Sq|K1) by RELAT_1:44; dom (Out_In_Sq|K1) c= dom ((proj2)*(Out_In_Sq|K1)) proof let x be set;assume A16:x in dom (Out_In_Sq|K1); then A17:x in dom Out_In_Sq /\ K1 by FUNCT_1:68; A18:(Out_In_Sq|K1).x=Out_In_Sq.x by A16,FUNCT_1:68; A19: dom proj2 = (the carrier of TOP-REAL 2) by FUNCT_2:def 1; x in dom Out_In_Sq by A17,XBOOLE_0:def 3; then Out_In_Sq.x in rng Out_In_Sq by FUNCT_1:12; then Out_In_Sq.x in the carrier of TOP-REAL 2 by XBOOLE_0:def 4; hence x in dom ((proj2)*(Out_In_Sq|K1)) by A16,A18,A19,FUNCT_1:21; end; then A20:dom ((proj2)*(Out_In_Sq|K1)) =dom (Out_In_Sq|K1) by A15,XBOOLE_0:def 10 .=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A13,FUNCT_2:def 1 .=K1 by A11,XBOOLE_1:28 .=[#]((TOP-REAL 2)|K1) by PRE_TOPC:def 10 .=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:12; rng ((proj2)*(Out_In_Sq|K1)) c= rng (proj2) by RELAT_1:45; then rng ((proj2)*(Out_In_Sq|K1)) c= the carrier of R^1 by TOPMETR:24,XBOOLE_1: 1; then (proj2)*(Out_In_Sq|K1) is Function of the carrier of (TOP-REAL 2)|K1, the carrier of R^1 by A20,FUNCT_2:4; then reconsider g1=(proj2)*(Out_In_Sq|K1) as map of (TOP-REAL 2)|K1,R^1; A21: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 proof let q be Point of TOP-REAL 2; assume A22:q in the carrier of (TOP-REAL 2)|K1; the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A23: q=p3 & ((p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2) & p3<>0.REAL 2) by A1,A22; now assume A24:q`2=0; then q`1=0 by A23; hence contradiction by A23,A24,EUCLID:57,58; end; hence q`2<>0; end; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g2.p=p`1/p`2/p`2 proof let p be Point of TOP-REAL 2; assume A25: p in the carrier of (TOP-REAL 2)|K1; A26: (the carrier of TOP-REAL 2)\{0.REAL 2}<>{} by Th19; A27:dom (Out_In_Sq|K1)=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A26,FUNCT_2:def 1 .=K1 by A11,XBOOLE_1:28; A28:the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A29: p=p3 & ((p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2) & p3<>0.REAL 2) by A1,A25; A30:Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A29,Th24; (Out_In_Sq|K1).p=Out_In_Sq.p by A25,A28,FUNCT_1:72; then g2.p=(proj1).(|[p`1/p`2/p`2,1/p`2]|) by A25,A27,A28,A30,FUNCT_1:23 .=(|[p`1/p`2/p`2,1/p`2]|)`1 by PSCOMP_1:def 28 .=p`1/p`2/p`2 by EUCLID:56; hence g2.p=p`1/p`2/p`2; end; then consider f2 being map of (TOP-REAL 2)|K1,R^1 such that A31:for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds f2.p=p`1/p`2/p`2; A32:f2 is continuous by A21,A31,Th44; for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds g1.p=1/p`2 proof let p be Point of TOP-REAL 2; assume A33: p in the carrier of (TOP-REAL 2)|K1; A34:K1 c= (the carrier of TOP-REAL 2)\{0.REAL 2} proof let z be set;assume z in K1; then consider p8 being Point of TOP-REAL 2 such that A35: p8=z &( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2) by A1; z in (the carrier of TOP-REAL 2) & not z in {0.REAL 2} by A35,TARSKI:def 1; hence z in (the carrier of TOP-REAL 2)\{0.REAL 2} by XBOOLE_0:def 4; end; A36: (the carrier of TOP-REAL 2)\{0.REAL 2}<>{} by Th19; A37:dom (Out_In_Sq|K1)=dom Out_In_Sq /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\{0.REAL 2})/\ K1 by A36,FUNCT_2:def 1 .=K1 by A34,XBOOLE_1:28; A38:the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A39: p=p3 & ((p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2) & p3<>0.REAL 2) by A1,A33; A40:Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A39,Th24; (Out_In_Sq|K1).p=Out_In_Sq.p by A33,A38,FUNCT_1:72; then g1.p=(proj2).(|[p`1/p`2/p`2,1/p`2]|) by A33,A37,A38,A40,FUNCT_1:23 .=(|[p`1/p`2/p`2,1/p`2]|)`2 by PSCOMP_1:def 29 .=1/p`2 by EUCLID:56; hence g1.p=1/p`2; end; then consider f1 being map of (TOP-REAL 2)|K1,R^1 such that A41:for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1 holds f1.p=1/p`2; A42:f1 is continuous by A21,A41,Th42; for x,y,s,r being real number st |[x,y]| in K1 & s=f2.(|[x,y]|) & r=f1.(|[x,y]|) holds f. |[x,y]|=|[s,r]| proof let x,y,s,r be real number; assume A43: |[x,y]| in K1 & s=f2.(|[x,y]|) & r=f1.(|[x,y]|); set p99=|[x,y]|; A44:the carrier of (TOP-REAL 2)|K1=[#]((TOP-REAL 2)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; consider p3 being Point of TOP-REAL 2 such that A45: p99=p3 & ((p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2) & p3<>0.REAL 2) by A1,A43; A46:f1.p99=1/p99`2 by A41,A43,A44; (Out_In_Sq|K0). |[x,y]|=(Out_In_Sq). |[x,y]| by A43,FUNCT_1:72 .= |[p99`1/p99`2/p99`2,1/p99`2]| by A45,Th24 .=|[s,r]| by A31,A43,A44,A46; hence f. |[x,y]|=|[s,r]| by A1; end; hence thesis by A2,A3,A32,A42,Th45; end; scheme TopSubset { P[set] } : { p where p is Point of TOP-REAL 2 : P[p] } is Subset of TOP-REAL 2 proof { p where p is Point of TOP-REAL 2 : P[p] } c= the carrier of TOP-REAL 2 proof let x be set; assume x in { p where p is Point of TOP-REAL 2 : P[p] }; then consider p being Point of TOP-REAL 2 such that A1: p = x & P[p]; thus x in the carrier of TOP-REAL 2 by A1; end; hence thesis; end; scheme TopCompl { P[set], K() -> Subset of TOP-REAL 2 } : K()` = { p where p is Point of TOP-REAL 2 : not P[p] } provided A1: K() = { p where p is Point of TOP-REAL 2 : P[p] } proof thus K()` c= { p where p is Point of TOP-REAL 2: not P[p] } proof let x be set;assume A2:x in K()`; then x in K()`; then x in (the carrier of TOP-REAL 2) \ K() by SUBSET_1:def 5; then A3:x in (the carrier of TOP-REAL 2) & not x in K() by XBOOLE_0:def 4; reconsider qx=x as Point of TOP-REAL 2 by A2; not P[qx] by A1,A3; hence x in {p7 where p7 is Point of TOP-REAL 2: not P[p7]}; end; let x be set;assume x in {p7 where p7 is Point of TOP-REAL 2: not P[p7]}; then consider p7 being Point of TOP-REAL 2 such that A4: p7=x & not P[p7]; not ex q7 being Point of TOP-REAL 2 st x=q7 & P[q7] by A4; then not x in K() by A1; then x in (the carrier of TOP-REAL 2) \ K() by A4,XBOOLE_0:def 4; then x in K()` by SUBSET_1:def 5; hence x in K()`; end; Lm2:now let p01, p02,px1,px2 be real number; set r0 = (p01 -p02)/4; assume p01 - px1 - (p02 - px2)<=r0--r0; then p01 - px1 - (p02 - px2)<=r0+r0 by XCMPLX_1:151; then p01 - px1 - p02 + px2<=r0+r0 by XCMPLX_1:37; then p01 - p02 - px1 + px2<=r0+r0 by XCMPLX_1:21; then p01 - p02 - (px1 - px2)<=r0+r0 by XCMPLX_1:37; then p01 - p02<= (px1 - px2)+(r0+r0) by REAL_1:86; then p01 - p02 - (r0+r0)<= (px1 - px2) by REAL_1:86; then p01 - p02 - (p01 -p02)/2<= (px1 - px2) by XCMPLX_1:72; then (p01 - p02)/2+(p01 - p02)/2 - (p01 -p02)/2<= (px1 - px2) by XCMPLX_1:66 ; hence (p01 - p02)/2<= px1 - px2 by XCMPLX_1:26; end; scheme ClosedSubset { F,G(Point of TOP-REAL 2) -> real number } : {p where p is Point of TOP-REAL 2 : F(p) <= G(p) } is closed Subset of TOP-REAL 2 provided A1: for p,q being Point of TOP-REAL 2 holds F(p-q) = F(p) - F(q) & G(p-q) = G(p) - G(q) and A2: for p,q being Point of TOP-REAL 2 holds (|. (p-q).|)^2 = (F(p-q))^2+(G(p-q))^2 proof defpred P[Point of TOP-REAL 2] means F($1) <= G($1); reconsider K2 = {p7 where p7 is Point of TOP-REAL 2: P[p7] } as Subset of TOP-REAL 2 from TopSubset; A3: K2 = {p7 where p7 is Point of TOP-REAL 2: P[p7] }; A4: K2`={p7 where p7 is Point of TOP-REAL 2:not P[p7]} from TopCompl(A3); for p being Point of Euclid 2 st p in K2` ex r being real number st r>0 & Ball(p,r) c= K2` proof let p be Point of Euclid 2; assume A5: p in K2`; then reconsider p0=p as Point of TOP-REAL 2; set r0=(F(p0) -G(p0))/4; consider p7 being Point of TOP-REAL 2 such that A6: p0=p7 & F(p7)>G(p7) by A4,A5; A7:F(p0)- G(p0)>0 by A6,SQUARE_1:11; then A8:r0>0 by REAL_2:127; A9: (F(p0) -G(p0))/2 >0 by A7,REAL_2:127; Ball(p,r0) c= K2` proof let x be set;assume A10: x in Ball(p,r0); then reconsider px=x as Point of TOP-REAL 2 by TOPREAL3:13; Ball(p,r0)={q where q is Element of Euclid 2: dist(p,q) < r0} by METRIC_1:18; then consider q being Element of Euclid 2 such that A11: q=x & dist(p,q) < r0 by A10; A12:dist(p,q)= |. (p0-px).| by A11,JGRAPH_1:45; A13:(|. (p0-px).|)^2 =(F(p0-px))^2+(G(p0-px))^2 by A2; A14:(G(p0-px))^2 >= 0 by SQUARE_1:72; (F(p0-px))^2 >= 0 by SQUARE_1:72; then A15:(G(p0-px))^2+0 <= (G(p0-px))^2 + (F(p0-px))^2 by REAL_1:55; A16: 0+(F(p0-px))^2 <= (G(p0-px))^2 + (F(p0-px))^2 by A14,REAL_1:55; 0 <= dist(p,q) by METRIC_1:5; then A17:(|.(p0-px).|)^2 <= r0^2 by A11,A12,SQUARE_1:77; then A18: (G(p0-px))^2 <= r0^2 by A13,A15,AXIOMS:22; A19: (F(p0-px))^2 <= r0^2 by A13,A16,A17,AXIOMS:22; A20:G(p0-px)=G(p0) - G(px) & F(p0-px)=F(p0) - F(px) by A1; then A21: -r0 <=G(p0) - G(px) & G(p0) - G(px)<=r0 by A8,A18,Th5; -r0 <=F(p0) - F(px) & F(p0) - F(px)<=r0 by A8,A19,A20,Th5; then F(p0) - F(px) - (G(p0) - G(px))<=r0--r0 by A21,REAL_1:92; then F(px)-G(px)>0 by A9,Lm2; then F(px)>G(px) by REAL_2:106; hence x in K2` by A4; end; hence ex r being real number st r>0 & Ball(p,r) c= K2` by A8; end; then K2` is open by Lm1,TOPMETR:22; hence thesis by TOPS_1:29; end; deffunc F(Point of TOP-REAL 2)=$1`1; deffunc G(Point of TOP-REAL 2)=$1`2; Lm3: for p,q being Point of TOP-REAL 2 holds F(p-q) = F(p) - F(q) & G(p-q) = G(p) - G(q) by TOPREAL3:8; Lm4: for p,q being Point of TOP-REAL 2 holds (|. (p-q).|)^2 = (F(p-q))^2+(G(p-q))^2 by JGRAPH_1:46; Lm5: {p7 where p7 is Point of TOP-REAL 2:F(p7)<=G(p7) } is closed Subset of TOP-REAL 2 from ClosedSubset(Lm3,Lm4); Lm6: for p,q being Point of TOP-REAL 2 holds G(p-q) = G(p) - G(q) & F(p-q) = F(p) - F(q) by TOPREAL3:8; Lm7: for p,q being Point of TOP-REAL 2 holds (|. (p-q).|)^2 = (G(p-q))^2+(F(p-q))^2 by JGRAPH_1:46; Lm8: {p7 where p7 is Point of TOP-REAL 2:G(p7)<=F(p7) } is closed Subset of TOP-REAL 2 from ClosedSubset(Lm6,Lm7); deffunc H(Point of TOP-REAL 2)=-$1`1; deffunc I(Point of TOP-REAL 2)=-$1`2; Lm9: now let p,q be Point of TOP-REAL 2; thus H(p-q) = -(p`1 - q`1) by TOPREAL3:8 .= -p`1 + q`1 by XCMPLX_1:162 .= H(p) - H(q) by XCMPLX_1:151; thus G(p-q) = G(p) - G(q) by TOPREAL3:8; end; Lm10: now let p,q be Point of TOP-REAL 2; (H(p-q))^2 = (F(p-q))^2 by SQUARE_1:61; hence (|. (p-q).|)^2 = (H(p-q))^2+(G(p-q))^2 by JGRAPH_1:46; end; Lm11: {p7 where p7 is Point of TOP-REAL 2:H(p7)<=G(p7) } is closed Subset of TOP-REAL 2 from ClosedSubset(Lm9,Lm10); Lm12: now let p,q be Point of TOP-REAL 2; thus G(p-q) = G(p) - G(q) by TOPREAL3:8; thus H(p-q) = -(p`1 - q`1) by TOPREAL3:8 .= -p`1 + q`1 by XCMPLX_1:162 .= H(p) - H(q) by XCMPLX_1:151; end; Lm13:now let p,q be Point of TOP-REAL 2; (-(p-q)`1)^2 = ((p-q)`1)^2 by SQUARE_1:61; hence (|. (p-q).|)^2 = (G(p-q))^2+(H(p-q))^2 by JGRAPH_1:46; end; Lm14: {p7 where p7 is Point of TOP-REAL 2:G(p7)<=H(p7) } is closed Subset of TOP-REAL 2 from ClosedSubset(Lm12,Lm13); Lm15: now let p,q be Point of TOP-REAL 2; thus I(p-q) = -(p`2 - q`2) by TOPREAL3:8 .= -p`2 + q`2 by XCMPLX_1:162 .= I(p) - I(q) by XCMPLX_1:151; thus F(p-q) = F(p) - F(q) by TOPREAL3:8; end; Lm16: now let p,q be Point of TOP-REAL 2; (-(p-q)`2)^2 = ((p-q)`2)^2 by SQUARE_1:61; hence (|. (p-q).|)^2 = (I(p-q))^2+(F(p-q))^2 by JGRAPH_1:46; end; Lm17: {p7 where p7 is Point of TOP-REAL 2:I(p7)<=F(p7) } is closed Subset of TOP-REAL 2 from ClosedSubset(Lm15,Lm16); Lm18:now let p,q be Point of TOP-REAL 2; thus F(p-q) = F(p) - F(q) by TOPREAL3:8; thus I(p-q) = -(p`2 - q`2) by TOPREAL3:8 .= -p`2 + q`2 by XCMPLX_1:162 .= I(p) - I(q) by XCMPLX_1:151; end; Lm19: now let p,q be Point of TOP-REAL 2; (I(p-q))^2 = (G(p-q))^2 by SQUARE_1:61; hence (|. (p-q).|)^2 = (F(p-q))^2+(I(p-q))^2 by JGRAPH_1:46; end; Lm20: {p7 where p7 is Point of TOP-REAL 2: F(p7)<=I(p7) } is closed Subset of TOP-REAL 2 from ClosedSubset(Lm18,Lm19); theorem Th48: for B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being map of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} holds f is continuous & K0 is closed proof let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0,f being map of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); assume A1: f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2}; the carrier of (TOP-REAL 2)|B0=[#]((TOP-REAL 2)|B0) by PRE_TOPC:12 .= B0 by PRE_TOPC:def 10; then K0 c= the carrier of TOP-REAL 2 by XBOOLE_1:1; then reconsider K1=K0 as Subset of TOP-REAL 2; K0 c= B0 proof let x be set;assume x in K0; then consider p8 being Point of TOP-REAL 2 such that A2: x=p8 & ( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2) by A1; not x in {0.REAL 2} by A2,TARSKI:def 1; hence x in B0 by A1,A2,XBOOLE_0:def 4; end; then A3: ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by JORDAN6:47; defpred P[Point of TOP-REAL 2] means ($1`2<=$1`1 & -$1`1<=$1`2 or $1`2>=$1`1 & $1`2<=-$1`1); reconsider K1={p7 where p7 is Point of TOP-REAL 2: P[p7]} as Subset of TOP-REAL 2 from TopSubset; reconsider K2={p7 where p7 is Point of TOP-REAL 2: p7`2<=p7`1 } as closed Subset of TOP-REAL 2 by Lm8; reconsider K3={p7 where p7 is Point of TOP-REAL 2: -p7`1<=p7`2 } as closed Subset of TOP-REAL 2 by Lm11; reconsider K4={p7 where p7 is Point of TOP-REAL 2: p7`1<=p7`2 } as closed Subset of TOP-REAL 2 by Lm5; reconsider K5={p7 where p7 is Point of TOP-REAL 2: p7`2<=-p7`1 } as closed Subset of TOP-REAL 2 by Lm14; A4:K2 /\ K3 is closed by TOPS_1:35; A5:K4 /\ K5 is closed by TOPS_1:35; A6:K2 /\ K3 \/ K4 /\ K5 c= K1 proof let x be set;assume A7:x in K2 /\ K3 \/ K4 /\ K5; now per cases by A7,XBOOLE_0:def 2; case x in K2 /\ K3; then A8:x in K2 & x in K3 by XBOOLE_0:def 3; then consider p7 being Point of TOP-REAL 2 such that A9: p7=x & p7`2<=(p7`1); consider p8 being Point of TOP-REAL 2 such that A10: p8=x & -p8`1<=p8`2 by A8; thus x in K1 by A9,A10; case x in K4 /\ K5; then A11:x in K4 & x in K5 by XBOOLE_0:def 3; then consider p7 being Point of TOP-REAL 2 such that A12: p7=x & p7`2>=(p7`1); consider p8 being Point of TOP-REAL 2 such that A13: p8=x & p8`2<= -p8`1 by A11; thus x in K1 by A12,A13; end; hence x in K1; end; K1 c= K2 /\ K3 \/ K4 /\ K5 proof let x be set;assume x in K1; then consider p being Point of TOP-REAL 2 such that A14: p=x & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1); x in K2 & x in K3 or x in K4 & x in K5 by A14; then x in K2 /\ K3 or x in K4 /\ K5 by XBOOLE_0:def 3; hence x in K2 /\ K3 \/ K4 /\ K5 by XBOOLE_0:def 2; end; then K1=K2 /\ K3 \/ K4 /\ K5 by A6,XBOOLE_0:def 10; then A15:K1 is closed by A4,A5,TOPS_1:36; A16:K1 /\ B0 c= K0 proof let x be set;assume x in K1 /\ B0; then A17:x in K1 & x in B0 by XBOOLE_0:def 3; then consider p7 being Point of TOP-REAL 2 such that A18: p7=x & (p7`2<=(p7`1) & -(p7`1)<=p7`2 or p7`2>=(p7`1) & p7`2<=-(p7`1)); x in the carrier of TOP-REAL 2 & not x in {0.REAL 2} by A1,A17,XBOOLE_0:def 4; then not x=0.REAL 2 by TARSKI:def 1; hence x in K0 by A1,A18; end; K0 c= K1 /\ B0 proof let x be set;assume x in K0; then consider p being Point of TOP-REAL 2 such that A19: x=p & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2 by A1; x in the carrier of TOP-REAL 2 & not x in {0.REAL 2} by A19,TARSKI:def 1; then x in K1 & x in B0 by A1,A19,XBOOLE_0:def 4; hence x in K1 /\ B0 by XBOOLE_0:def 3; end; then K0=K1 /\ B0 by A16,XBOOLE_0:def 10 .=K1 /\ [#]((TOP-REAL 2)|B0) by PRE_TOPC:def 10; hence thesis by A1,A3,A15,Th46,PRE_TOPC:43; end; theorem Th49: for B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2)|B0,f being map of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} holds f is continuous & K0 is closed proof let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0,f being map of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0); assume A1: f=Out_In_Sq|K0 & B0=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2}; the carrier of (TOP-REAL 2)|B0=[#]((TOP-REAL 2)|B0) by PRE_TOPC:12 .= B0 by PRE_TOPC:def 10; then K0 c= the carrier of TOP-REAL 2 by XBOOLE_1:1; then reconsider K1=K0 as Subset of TOP-REAL 2; K0 c= B0 proof let x be set;assume x in K0; then consider p8 being Point of TOP-REAL 2 such that A2: x=p8 & ( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2) by A1; x in the carrier of TOP-REAL 2 & not x in {0.REAL 2} by A2,TARSKI:def 1; hence x in B0 by A1,XBOOLE_0:def 4; end; then A3: ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by JORDAN6:47; defpred P[Point of TOP-REAL 2] means ($1`1<=$1`2 & -$1`2<=$1`1 or $1`1>=$1`2 & $1`1<=-$1`2); reconsider K1={p7 where p7 is Point of TOP-REAL 2: P[p7]} as Subset of TOP-REAL 2 from TopSubset; reconsider K2={p7 where p7 is Point of TOP-REAL 2: p7`1<=p7`2 } as closed Subset of TOP-REAL 2 by Lm5; reconsider K3={p7 where p7 is Point of TOP-REAL 2: -p7`2<=p7`1 } as closed Subset of TOP-REAL 2 by Lm17; reconsider K4={p7 where p7 is Point of TOP-REAL 2: p7`2<=p7`1 } as closed Subset of TOP-REAL 2 by Lm8; reconsider K5={p7 where p7 is Point of TOP-REAL 2: p7`1<=-p7`2 } as closed Subset of TOP-REAL 2 by Lm20; A4:K2 /\ K3 is closed by TOPS_1:35; A5:K4 /\ K5 is closed by TOPS_1:35; A6:K2 /\ K3 \/ K4 /\ K5 c= K1 proof let x be set;assume A7:x in K2 /\ K3 \/ K4 /\ K5; now per cases by A7,XBOOLE_0:def 2; case x in K2 /\ K3; then A8:x in K2 & x in K3 by XBOOLE_0:def 3; then consider p7 being Point of TOP-REAL 2 such that A9: p7=x & p7`1<=(p7`2); consider p8 being Point of TOP-REAL 2 such that A10: p8=x & -p8`2<=p8`1 by A8; thus x in K1 by A9,A10; case x in K4 /\ K5; then A11:x in K4 & x in K5 by XBOOLE_0:def 3; then consider p7 being Point of TOP-REAL 2 such that A12: p7=x & p7`1>=(p7`2); consider p8 being Point of TOP-REAL 2 such that A13: p8=x & p8`1<= -p8`2 by A11; thus x in K1 by A12,A13; end; hence x in K1; end; K1 c= K2 /\ K3 \/ K4 /\ K5 proof let x be set;assume x in K1; then consider p being Point of TOP-REAL 2 such that A14: p=x & (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2); x in K2 & x in K3 or x in K4 & x in K5 by A14; then x in K2 /\ K3 or x in K4 /\ K5 by XBOOLE_0:def 3; hence x in K2 /\ K3 \/ K4 /\ K5 by XBOOLE_0:def 2; end; then K1=K2 /\ K3 \/ K4 /\ K5 by A6,XBOOLE_0:def 10; then A15:K1 is closed by A4,A5,TOPS_1:36; A16:K1 /\ B0 c= K0 proof let x be set;assume x in K1 /\ B0; then A17:x in K1 & x in B0 by XBOOLE_0:def 3; then consider p7 being Point of TOP-REAL 2 such that A18: p7=x & (p7`1<=(p7`2) & -(p7`2)<=p7`1 or p7`1>=(p7`2) & p7`1<=-(p7`2)); x in the carrier of TOP-REAL 2 & not x in {0.REAL 2} by A1,A17,XBOOLE_0:def 4; then not x=0.REAL 2 by TARSKI:def 1; hence x in K0 by A1,A18; end; K0 c= K1 /\ B0 proof let x be set;assume x in K0; then consider p being Point of TOP-REAL 2 such that A19: x=p & (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2 by A1; x in the carrier of TOP-REAL 2 & not x in {0.REAL 2} by A19,TARSKI:def 1; then x in K1 & x in B0 by A1,A19,XBOOLE_0:def 4; hence x in K1 /\ B0 by XBOOLE_0:def 3; end; then K0=K1 /\ B0 by A16,XBOOLE_0:def 10 .=K1 /\ [#]((TOP-REAL 2)|B0) by PRE_TOPC:def 10; hence thesis by A1,A3,A15,Th47,PRE_TOPC:43; end; theorem Th50:for D being non empty Subset of TOP-REAL 2 st D`={0.REAL 2} holds ex h being map of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=Out_In_Sq & h is continuous proof let D be non empty Subset of TOP-REAL 2; assume A1:D`={0.REAL 2}; reconsider B0= {0.REAL 2} as Subset of TOP-REAL 2; A2: D=(B0)` by A1 .=((the carrier of TOP-REAL 2)\ {0.REAL 2}) by SUBSET_1:def 5; A3:{p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} c= the carrier of (TOP-REAL 2)|D proof let x be set; assume x in {p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2}; then consider p such that A4: x=p & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.REAL 2; now assume not x in D; then x in (the carrier of TOP-REAL 2) \ D by A4,XBOOLE_0:def 4; then x in D` by SUBSET_1:def 5; hence contradiction by A1,A4,TARSKI:def 1; end; then x in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 10; hence x in the carrier of (TOP-REAL 2)|D; end; ((1.REAL 2)`2<=(1.REAL 2)`1 & -(1.REAL 2)`1<=(1.REAL 2)`2 or (1.REAL 2)`2>=(1.REAL 2)`1 & (1.REAL 2)`2<=-(1.REAL 2)`1) & (1.REAL 2)<>0.REAL 2 by Th13,REVROT_1:19; then 1.REAL 2 in {p where p is Point of TOP-REAL 2: (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2}; then reconsider K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} as non empty Subset of (TOP-REAL 2)|D by A3; A5:{p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} c= the carrier of (TOP-REAL 2)|D proof let x be set; assume x in {p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2}; then consider p such that A6: x=p & (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2)& p<>0.REAL 2; now assume not x in D; then x in (the carrier of TOP-REAL 2) \ D by A6,XBOOLE_0:def 4; then x in D` by SUBSET_1:def 5; hence contradiction by A1,A6,TARSKI:def 1; end; then x in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 10; hence x in the carrier of (TOP-REAL 2)|D; end; set Y1=|[-1,1]|; Y1`1=-1 & Y1`2=1 by EUCLID:56; then Y1 in {p where p is Point of TOP-REAL 2: (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} by Th11; then reconsider K1={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} as non empty Subset of (TOP-REAL 2)|D by A5; A7:the carrier of ((TOP-REAL 2)|D) =[#]((TOP-REAL 2)|D) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; A8:K0 c= (the carrier of TOP-REAL 2)\{0.REAL 2} proof let z be set;assume z in K0; then consider p8 being Point of TOP-REAL 2 such that A9: p8=z &( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2); z in (the carrier of TOP-REAL 2) & not z in {0.REAL 2} by A9,TARSKI:def 1; hence z in (the carrier of TOP-REAL 2)\{0.REAL 2} by XBOOLE_0:def 4; end; A10:(the carrier of TOP-REAL 2)\ {0.REAL 2}<> {} by Th19; A11:dom (Out_In_Sq|K0)= dom (Out_In_Sq) /\ K0 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\ {0.REAL 2}) /\ K0 by A10,FUNCT_2:def 1 .=K0 by A8,XBOOLE_1:28; A12: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 10 .=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:12; A13:the carrier of ((TOP-REAL 2)|D)|K0 =[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10 .=((the carrier of TOP-REAL 2)\ {0.REAL 2}) /\ K0 by A8,XBOOLE_1:28; the carrier of ((TOP-REAL 2)|D)=[#](((TOP-REAL 2)|D)) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; then A14:the carrier of ((TOP-REAL 2)|D)|K0 c= the carrier of ((TOP-REAL 2)|D) by A2,A13,XBOOLE_1:17; rng (Out_In_Sq|K0) c= the carrier of ((TOP-REAL 2)|D)|K0 proof let y be set;assume y in rng (Out_In_Sq|K0); then consider x being set such that A15:x in dom (Out_In_Sq|K0) & y=(Out_In_Sq|K0).x by FUNCT_1:def 5; A16:x in (dom Out_In_Sq) /\ K0 by A15,FUNCT_1:68; then A17:x in K0 by XBOOLE_0:def 3; A18: K0 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A17; A19:Out_In_Sq.p=y by A15,A17,FUNCT_1:72; consider px being Point of TOP-REAL 2 such that A20: x=px & (px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1) & px<>0.REAL 2 by A17; reconsider K00=K0 as Subset of TOP-REAL 2 by A18; K00=[#]((TOP-REAL 2)|K00) by PRE_TOPC:def 10 .=the carrier of ((TOP-REAL 2)|K00) by PRE_TOPC:12; then A21:p in the carrier of ((TOP-REAL 2)|K00) by A16,XBOOLE_0:def 3; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K00 holds q`1<>0 proof let q be Point of TOP-REAL 2; assume A22:q in the carrier of (TOP-REAL 2)|K00; the carrier of (TOP-REAL 2)|K00=[#]((TOP-REAL 2)|K00) by PRE_TOPC:12 .=K0 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A23: q=p3 & ((p3`2<=p3`1 & -p3`1<=p3`2 or p3`2>=p3`1 & p3`2<=-p3`1) & p3<>0.REAL 2) by A22; now assume A24:q`1=0; then q`2=0 by A23; hence contradiction by A23,A24,EUCLID:57,58; end; hence q`1<>0; end; then A25:p`1<>0 by A21; set p9=|[1/p`1,p`2/p`1/p`1]|; A26:p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:56; A27:now assume p9=0.REAL 2; then 0 *p`1=1/p`1*p`1 by A26,EUCLID:56,58; hence contradiction by A25,XCMPLX_1:88; end; A28:Out_In_Sq.p=|[1/p`1,p`2/p`1/p`1]| by A20,Def1; now per cases; case A29: p`1>=0; then p`2/p`1<=p`1/p`1 & (-1 *p`1)/p`1<=p`2/p`1 or p`2>=p`1 & p`2<=-1 *p `1 by A20,A25,REAL_1:73; then p`2/p`1<=1 & (-1)*p`1/p`1<=p`2/p`1 or p`2>=p`1 & p`2<=-1 *p`1 by A25,XCMPLX_1:60,175; then A30: p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=p`1/p`1 & p`2<=-1 *p`1 by A25,A29,REAL_1:73,XCMPLX_1:90; then p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=1 & p`2<=(-1)*p`1 by A25,XCMPLX_1:60,175; then p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=1 & p`2/p`1<=(-1)*p`1/p`1 by A25,A29,REAL_1:73; then A31:p`2/p`1<=1 & -1<=p`2/p`1 or p`2/p`1>=1 & p`2/p`1<=-1 by A25,XCMPLX_1:90; not (p`2/p`1>=1 & p`2/p`1<=-1) by AXIOMS:22; then (-1)/p`1<= p`2/p`1/p`1 by A25,A29,A30,REAL_1:73,XCMPLX_1:60; then A32:p`2/p`1/p`1 <=1/p`1 & -(1/p`1)<= p`2/p`1/p`1 or p`2/p`1/p`1 >=1/p`1 & p`2/p`1/p`1<= -(1/p`1) by A25,A29,A31,AXIOMS:22, REAL_1:73,XCMPLX_1:188; p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:56; hence y in K0 by A19,A27,A28,A32; case A33:p`1<0; then p`2<=p`1 & (-1 *p`1)<=p`2 or p`2/p`1<=p`1/p`1 & p`2/p`1>=(-1 *p`1)/ p`1 by A20,REAL_1:74; then p`2<=p`1 & (-1 *p`1)<=p`2 or p`2/p`1<=1 & p`2/p`1>=(-1)*p`1/p`1 by A33,XCMPLX_1:60,175; then A34: p`2/p`1>=p`1/p`1 & (-1 *p`1)<=p`2 or p`2/p`1<=1 & p`2/p`1>=-1 by A33,REAL_1:74,XCMPLX_1:90; then p`2/p`1>=1 & (-1)*p`1<=p`2 or p`2/p`1<=1 & p`2/p`1>=-1 by A33,XCMPLX_1:60,175; then p`2/p`1>=1 & (-1)*p`1/p`1>=p`2/p`1 or p`2/p`1<=1 & p`2/p`1>=-1 by A33,REAL_1:74; then A35:p`2/p`1>=1 & -1>=p`2/p`1 or p`2/p`1<=1 & p`2/p`1>=-1 by A33,XCMPLX_1:90; not(p`2/p`1>=1 & p`2/p`1<=-1) by AXIOMS:22; then (-1)/p`1>= p`2/p`1/p`1 by A33,A34,REAL_1:74,XCMPLX_1:60; then A36:p`2/p`1/p`1 <=1/p`1 & -(1/p`1)<= p`2/p`1/p`1 or p`2/p`1/p`1 >=1/p`1 & p`2/p`1/p`1<= -(1/p`1) by A33,A35,AXIOMS:22,REAL_1: 74,XCMPLX_1:188; p9`1=1/p`1 & p9`2=p`2/p`1/p`1 by EUCLID:56; hence y in K0 by A19,A27,A28,A36; end; then y in [#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 10; hence y in the carrier of ((TOP-REAL 2)|D)|K0; end; then rng (Out_In_Sq|K0)c= the carrier of ((TOP-REAL 2)|D) by A14,XBOOLE_1:1; then Out_In_Sq|K0 is Function of the carrier of ((TOP-REAL 2)|D)|K0, the carrier of ((TOP-REAL 2)|D) by A11,A12,FUNCT_2:4; then reconsider f=Out_In_Sq|K0 as map of ((TOP-REAL 2)|D)|K0,((TOP-REAL 2)|D); A37:K1 c= (the carrier of TOP-REAL 2)\{0.REAL 2} proof let z be set;assume z in K1; then consider p8 being Point of TOP-REAL 2 such that A38: p8=z &( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2); z in (the carrier of TOP-REAL 2) & not z in {0.REAL 2} by A38,TARSKI:def 1; hence z in (the carrier of TOP-REAL 2)\{0.REAL 2} by XBOOLE_0:def 4; end; A39:(the carrier of TOP-REAL 2)\ {0.REAL 2}<> {} by Th19; A40:dom (Out_In_Sq|K1)= dom (Out_In_Sq) /\ K1 by FUNCT_1:68 .=((the carrier of TOP-REAL 2)\ {0.REAL 2}) /\ K1 by A39,FUNCT_2:def 1 .=K1 by A37,XBOOLE_1:28; A41: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 10 .=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:12; A42:the carrier of ((TOP-REAL 2)|D)|K1 =[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10 .=((the carrier of TOP-REAL 2)\ {0.REAL 2}) /\ K1 by A37,XBOOLE_1:28; the carrier of ((TOP-REAL 2)|D)=[#](((TOP-REAL 2)|D)) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; then A43:the carrier of ((TOP-REAL 2)|D)|K1 c= the carrier of ((TOP-REAL 2)|D) by A2,A42,XBOOLE_1:17; rng (Out_In_Sq|K1) c= the carrier of ((TOP-REAL 2)|D)|K1 proof let y be set;assume y in rng (Out_In_Sq|K1); then consider x being set such that A44:x in dom (Out_In_Sq|K1) & y=(Out_In_Sq|K1).x by FUNCT_1:def 5; A45:x in (dom Out_In_Sq) /\ K1 by A44,FUNCT_1:68; then A46:x in K1 by XBOOLE_0:def 3; A47: K1 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1; then reconsider p=x as Point of TOP-REAL 2 by A46; A48:Out_In_Sq.p=y by A44,A46,FUNCT_1:72; consider px being Point of TOP-REAL 2 such that A49: x=px & (px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2) & px<>0.REAL 2 by A46; reconsider K10=K1 as Subset of TOP-REAL 2 by A47; K10=[#]((TOP-REAL 2)|K10) by PRE_TOPC:def 10 .=the carrier of ((TOP-REAL 2)|K10) by PRE_TOPC:12; then A50:p in the carrier of ((TOP-REAL 2)|K10) by A45,XBOOLE_0:def 3; for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K10 holds q`2<>0 proof let q be Point of TOP-REAL 2; assume A51:q in the carrier of (TOP-REAL 2)|K10; the carrier of (TOP-REAL 2)|K10=[#]((TOP-REAL 2)|K10) by PRE_TOPC:12 .=K1 by PRE_TOPC:def 10; then consider p3 being Point of TOP-REAL 2 such that A52: q=p3 & ((p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1<=-p3`2) & p3<>0.REAL 2) by A51; now assume A53:q`2=0; then q`1=0 by A52; hence contradiction by A52,A53,EUCLID:57,58; end; hence q`2<>0; end; then A54:p`2<>0 by A50; set p9=|[p`1/p`2/p`2,1/p`2]|; A55:now assume p9=0.REAL 2; then p9`2=0 & p9`1=0 by EUCLID:56,58; then 0 *p`2=1/p`2*p`2 by EUCLID:56; hence contradiction by A54,XCMPLX_1:88; end; A56:Out_In_Sq.p=|[p`1/p`2/p`2,1/p`2]| by A49,Th24; now per cases; case A57: p`2>=0; then p`1/p`2<=p`2/p`2 & (-1 *p`2)/p`2<=p`1/p`2 or p`1>=p`2 & p`1<=-1 *p `2 by A49,A54,REAL_1:73; then p`1/p`2<=1 & (-1)*p`2/p`2<=p`1/p`2 or p`1>=p`2 & p`1<=-1 *p`2 by A54,XCMPLX_1:60,175; then A58: p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=p`2/p`2 & p`1<=-1 *p`2 by A54,A57,REAL_1:73,XCMPLX_1:90; then p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=1 & p`1<=(-1)*p`2 by A54,XCMPLX_1:60,175; then p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=1 & p`1/p`2<=(-1)*p`2/p`2 by A54,A57,REAL_1:73; then A59:p`1/p`2<=1 & -1<=p`1/p`2 or p`1/p`2>=1 & p`1/p`2<=-1 by A54,XCMPLX_1:90; not(p`1/p`2>=1 & p`1/p`2<=-1) by AXIOMS:22; then (-1)/p`2<= p`1/p`2/p`2 by A54,A57,A58,REAL_1:73,XCMPLX_1:60; then A60:p`1/p`2/p`2 <=1/p`2 & -(1/p`2)<= p`1/p`2/p`2 or p`1/p`2/p`2 >=1/p`2 & p`1/p`2/p`2<= -(1/p`2) by A54,A57,A59,AXIOMS:22, REAL_1:73,XCMPLX_1:188; p9`2=1/p`2 & p9`1=p`1/p`2/p`2 by EUCLID:56; hence y in K1 by A48,A55,A56,A60; case A61:p`2<0; then p`1<=p`2 & (-1 *p`2)<=p`1 or p`1/p`2<=p`2/p`2 & p`1/p`2>=(-1 *p`2)/ p`2 by A49,REAL_1:74; then p`1<=p`2 & (-1 *p`2)<=p`1 or p`1/p`2<=1 & p`1/p`2>=(-1)*p`2/p`2 by A61,XCMPLX_1:60,175; then p`1/p`2>=p`2/p`2 & (-1 *p`2)<=p`1 or p`1/p`2<=1 & p`1/p`2>=-1 by A61,REAL_1:74,XCMPLX_1:90; then p`1/p`2>=1 & (-1)*p`2<=p`1 or p`1/p`2<=1 & p`1/p`2>=-1 by A61,XCMPLX_1:60,175; then p`1/p`2>=1 & (-1)*p`2/p`2>=p`1/p`2 or p`1/p`2<=1 & p`1/p`2>=-1 by A61,REAL_1:74; then A62:p`1/p`2>=1 & -1>=p`1/p`2 or p`1/p`2<=1 & p`1/p`2>=-1 by A61,XCMPLX_1:90; then (-1)/p`2>= p`1/p`2/p`2 by A61,AXIOMS:22,REAL_1:74; then A63:p`1/p`2/p`2 <=1/p`2 & -(1/p`2)<= p`1/p`2/p`2 or p`1/p`2/p`2 >=1/p`2 & p`1/p`2/p`2<= -(1/p`2) by A61,A62,AXIOMS:22,REAL_1: 74,XCMPLX_1:188; p9`2=1/p`2 & p9`1=p`1/p`2/p`2 by EUCLID:56; hence y in K1 by A48,A55,A56,A63; end; then y in [#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 10; hence y in the carrier of ((TOP-REAL 2)|D)|K1; end; then rng (Out_In_Sq|K1)c= the carrier of ((TOP-REAL 2)|D) by A43,XBOOLE_1:1; then Out_In_Sq|K1 is Function of the carrier of ((TOP-REAL 2)|D)|K1, the carrier of ((TOP-REAL 2)|D) by A40,A41,FUNCT_2:4; then reconsider g=Out_In_Sq|K1 as map of ((TOP-REAL 2)|D)|K1, ((TOP-REAL 2)|D); A64:K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 10; A65:K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 10; A66:D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 10; A67:K0 \/ K1 c= D proof let x be set;assume A68: x in K0 \/ K1; now per cases by A68,XBOOLE_0:def 2; case x in K0; then consider p such that A69:p=x & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2; thus x in the carrier of TOP-REAL 2 & not x=0.REAL 2 by A69; case x in K1; then consider p such that A70:p=x & (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2; thus x in the carrier of TOP-REAL 2 & not x=0.REAL 2 by A70; end; then x in the carrier of TOP-REAL 2 & not x in {0.REAL 2} by TARSKI:def 1 ; hence x in D by A2,XBOOLE_0:def 4; end; D c= K0 \/ K1 proof let x be set;assume A71: x in D; then A72:x in (the carrier of TOP-REAL 2) & not x in {0.REAL 2} by A2,XBOOLE_0:def 4; reconsider px=x as Point of TOP-REAL 2 by A71; (px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1) & px<>0.REAL 2 or (px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2) & px<>0.REAL 2 by A72,REAL_2:110,TARSKI:def 1; then x in K0 or x in K1; hence x in K0 \/ K1 by XBOOLE_0:def 2; end; then A73:([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#]((TOP-REAL 2)|D) by A64,A65,A66,A67,XBOOLE_0:def 10; f=Out_In_Sq|K0 & D=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.REAL 2} by A2; then A74: f is continuous & K0 is closed by Th48; g=Out_In_Sq|K1 & D=(the carrier of TOP-REAL 2) \ {0.REAL 2} & K1={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.REAL 2} by A2; then A75: g is continuous & K1 is closed by Th49; A76: for x be set st x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1))) holds f.x = g.x proof let x be set;assume x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1))); then A77:x in K0 & x in K1 by A64,A65,XBOOLE_0:def 3; then f.x=Out_In_Sq.x by FUNCT_1:72; hence f.x = g.x by A77,FUNCT_1:72; end; then consider h being map of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that A78: h= f+*g & h is continuous by A64,A65,A73,A74,A75,Th9; A79:dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1; A80:the carrier of ((TOP-REAL 2)|D) =[#](((TOP-REAL 2)|D)) by PRE_TOPC:12 .=((the carrier of TOP-REAL 2)\ {0.REAL 2}) by A2,PRE_TOPC:def 10; then A81:dom Out_In_Sq=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1; A82:dom f=K0 by A12,FUNCT_2:def 1; A83:K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 10; A84:dom g=K1 by A41,FUNCT_2:def 1; K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 10; then A85:f tolerates g by A76,A82,A83,A84,PARTFUN1:def 6; for x being set st x in dom h holds h.x=Out_In_Sq.x proof let x be set;assume A86: x in dom h; then x in (the carrier of TOP-REAL 2) & not x in {0.REAL 2} by A80,XBOOLE_0:def 4; then A87:x <>0.REAL 2 by TARSKI:def 1; reconsider p=x as Point of TOP-REAL 2 by A80,A86,XBOOLE_0:def 4; now per cases; case A88:x in K0; h.p=(g+*f).p by A78,A85,FUNCT_4:35 .=f.p by A82,A88,FUNCT_4:14; hence h.x=Out_In_Sq.x by A88,FUNCT_1:72; case not x in K0; then not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) by A87; then (p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2)by REAL_2:110; then A89:x in K1 by A87; then Out_In_Sq.p=g.p by FUNCT_1:72; hence h.x=Out_In_Sq.x by A78,A84,A89,FUNCT_4:14; end; hence h.x=Out_In_Sq.x; end; then f+*g=Out_In_Sq by A78,A79,A81,FUNCT_1:9; hence thesis by A64,A65,A73,A74,A75,A76,Th9; end; theorem Th51: for B,K0,Kb being Subset of TOP-REAL 2 st B={0.REAL 2} & K0={p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} & Kb={q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1} ex f being map of (TOP-REAL 2)|B`,(TOP-REAL 2)|B` st f is continuous & f is one-to-one & (for t being Point of TOP-REAL 2 st t in K0 & t<>0.REAL 2 holds not f.t in K0 \/ Kb) &(for r being Point of TOP-REAL 2 st not r in K0 \/ Kb holds f.r in K0) &(for s being Point of TOP-REAL 2 st s in Kb holds f.s=s) proof let B,K0,Kb be Subset of TOP-REAL 2; assume A1:B={0.REAL 2} & K0={p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} & Kb={q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1}; then A2:B`=(the carrier of TOP-REAL 2) \ {0.REAL 2} by SUBSET_1:def 5; reconsider D=B` as non empty Subset of TOP-REAL 2 by A1,Th19; A3:D`={0.REAL 2} by A1; then consider h being map of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that A4: h=Out_In_Sq & h is continuous by Th50; A5: D =((the carrier of TOP-REAL 2)\ {0.REAL 2}) by A1,SUBSET_1:def 5; set K0a={p8 where p8 is Point of TOP-REAL 2: (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2}; set K1a={p8 where p8 is Point of TOP-REAL 2: (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2 }; for x1,x2 being set st x1 in dom Out_In_Sq & x2 in dom Out_In_Sq & Out_In_Sq.x1=Out_In_Sq.x2 holds x1=x2 proof let x1,x2 be set;assume that A6: x1 in dom Out_In_Sq & x2 in dom Out_In_Sq and A7: Out_In_Sq.x1=Out_In_Sq.x2; (the carrier of TOP-REAL 2) \ {0.REAL 2}<>{} by Th19; then A8:dom Out_In_Sq=(the carrier of TOP-REAL 2) \ {0.REAL 2} by FUNCT_2:def 1; then reconsider p1=x1,p2=x2 as Point of TOP-REAL 2 by A6,XBOOLE_0:def 4; A9:the carrier of (TOP-REAL 2)|D=[#]((TOP-REAL 2)|D) by PRE_TOPC:12 .=D by PRE_TOPC:def 10; reconsider K01=K0a as non empty Subset of ((TOP-REAL 2)|D) by A3,Th27; ((1.REAL 2)`1<=(1.REAL 2)`2 & -(1.REAL 2)`2<=(1.REAL 2)`1 or (1.REAL 2)`1>=(1.REAL 2)`2 & (1.REAL 2)`1<=-(1.REAL 2)`2) & (1.REAL 2)<>0.REAL 2 by Th13,REVROT_1:19; then A10: 1.REAL 2 in K1a; K1a c= D proof let x be set;assume x in K1a; then consider p8 being Point of TOP-REAL 2 such that A11: x=p8 & ( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2); not x in {0.REAL 2} by A11,TARSKI:def 1; hence x in D by A2,A11,XBOOLE_0:def 4; end; then reconsider K11=K1a as non empty Subset of ((TOP-REAL 2)|D) by A9,A10; A12: D c= K01 \/ K11 proof let x be set;assume A13:x in D; then A14: x in (the carrier of TOP-REAL 2) & not x in {0.REAL 2} by A5,XBOOLE_0:def 4; reconsider px=x as Point of TOP-REAL 2 by A13; (px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1) & px<>0.REAL 2 or (px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2) & px<>0.REAL 2 by A14,REAL_2:109,TARSKI:def 1; then x in K01 or x in K11; hence x in K01 \/ K11 by XBOOLE_0:def 2; end; A15:x1 in D & x2 in D by A1,A6,A8,SUBSET_1:def 5; now per cases by A12,A15,XBOOLE_0:def 2; case x1 in K01; then consider p7 being Point of TOP-REAL 2 such that A16: p1=p7 & ( (p7`2<=p7`1 & -p7`1<=p7`2 or p7`2>=p7`1 & p7`2<=-p7`1) & p7<>0.REAL 2); A17:Out_In_Sq.p1=|[1/p1`1,p1`2/p1`1/p1`1]| by A16,Def1; now per cases by A12,A15,XBOOLE_0:def 2; case x2 in K0a; then consider p8 being Point of (TOP-REAL 2) such that A18: p2=p8 & ( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2); A19: |[1/p2`1,p2`2/p2`1/p2`1]| =|[1/p1`1,p1`2/p1`1/p1`1]| by A7,A17,A18,Def1; set qq=|[1/p2`1,p2`2/p2`1/p2`1]|; qq`1=1/p2`1 & qq`2=p2`2/p2`1/p2`1 by EUCLID:56; then A20:1/p1`1= 1/p2`1 & p1`2/p1`1/p1`1 = p2`2/p2`1/p2`1 by A19,EUCLID:56; A21:(1/p1`1)"=(p1`1)"" by XCMPLX_1:217 .=p1`1; A22: (1/p2`1)"=(p2`1)"" by XCMPLX_1:217 .=p2`1; A23:now assume A24:p1`1=0; then p1`2=0 by A16; hence contradiction by A16,A24,EUCLID:57,58; end; then p1`2/p1`1= p2`2/p1`1 by A20,A21,A22,XCMPLX_1:53; then A25:p1`2=p2`2 by A23,XCMPLX_1:53; p1=|[p1`1,p1`2]| by EUCLID:57; hence x1=x2 by A20,A21,A22,A25,EUCLID:57; case A26:x2 in K1a & not x2 in K0a; then consider p8 being Point of (TOP-REAL 2) such that A27: p2=p8 & ((p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2); not((p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2<=-p2`1) & p2 <> 0.REAL 2) by A26; then Out_In_Sq.p2=|[p2`1/p2`2/p2`2,1/p2`2]| by A27,Def1; then A28:1/p1`1=p2`1/p2`2/p2`2 & p1`2/p1`1/p1`1=1/p2`2 by A7,A17,SPPOL_2:1; A29:now assume A30:p2`2=0; then p2`1=0 by A27; hence contradiction by A27,A30,EUCLID:57,58; end; A31:now assume A32:p1`1=0; then p1`2=0 by A16; hence contradiction by A16,A32,EUCLID:57,58; end; A33:p2`1/p2`2=1/p1`1*p2`2 by A28,A29,XCMPLX_1:88 .= p2`2/p1`1 by XCMPLX_1:100; A34:p1`2/p1`1=1/p2`2*p1`1 by A28,A31,XCMPLX_1:88 .= p1`1/p2`2 by XCMPLX_1:100; then A35:(p2`1/p2`2)* (p1`2/p1`1)=1 by A29,A31,A33,XCMPLX_1:113; A36: (p2`1/p2`2)* (p1`2/p1`1)*p1`1=1 *p1`1 by A29,A31,A33,A34,XCMPLX_1 :113; then (p2`1/p2`2)* ((p1`2/p1`1)*p1`1)=p1`1 by XCMPLX_1:4; then A37:(p2`1/p2`2)*p1`2=p1`1 by A31,XCMPLX_1:88; A38:p2`1<>0 & p1`2<>0 by A31,A36; then A39:(p2`1/p2`2)=p1`1/p1`2 by A37,XCMPLX_1:90; consider p9 being Point of (TOP-REAL 2) such that A40: p2=p9 & ( (p9`1<=p9`2 & -p9`2<=p9`1 or p9`1>=p9`2 & p9`1<=-p9`2) & p9<>0.REAL 2) by A26; A41:now per cases by A40; case A42:p2`1<=p2`2 & -p2`2<=p2`1; now assume A43:p2`2<0; then 0<p2`1 by A42,REAL_1:66; hence contradiction by A42,A43,AXIOMS:22; end; then p2`1/p2`2<=p2`2/p2`2 by A29,A42,REAL_1:73; hence p2`1/p2`2<=1 by A29,XCMPLX_1:60; case A44:p2`1>=p2`2 & p2`1<=-p2`2; now assume A45:p2`2>0; then -p2`2< -0 by REAL_1:50; then 0>p2`1 by A44; hence contradiction by A44,A45,AXIOMS:22; end; then p2`1/p2`2<=p2`2/p2`2 by A29,A44,REAL_1:74; hence p2`1/p2`2<=1 by A29,XCMPLX_1:60; end; A46:now per cases by A16; case A47:p1`2<=p1`1 & -p1`1<=p1`2; now assume A48:p1`1<0; then 0<p1`2 by A47,REAL_1:66; hence contradiction by A47,A48,AXIOMS:22; end; then p1`2/p1`1<=p1`1/p1`1 by A31,A47,REAL_1:73; hence p1`2/p1`1<=1 by A31,XCMPLX_1:60; case A49:p1`2>=p1`1 & p1`2<=-p1`1; now assume A50:p1`1>0; then -p1`1< -0 by REAL_1:50; then 0>p1`2 by A49; hence contradiction by A49,A50,AXIOMS:22; end; then p1`2/p1`1<=p1`1/p1`1 by A31,A49,REAL_1:74; hence p1`2/p1`1<=1 by A31,XCMPLX_1:60; end; A51:now per cases by A40; case A52:p2`1<=p2`2 & -p2`2<=p2`1; now assume A53:p2`2<0; then 0<p2`1 by A52,REAL_1:66; hence contradiction by A52,A53,AXIOMS:22; end; then (-p2`2)/p2`2<=p2`1/p2`2 by A29,A52,REAL_1:73; hence -1<=p2`1/p2`2 by A29,XCMPLX_1:198; case A54:p2`1>=p2`2 & p2`1<=-p2`2; then A55: -p2`1>=--p2`2 by REAL_1:50; now assume A56:p2`2>0; then -p2`2< -0 by REAL_1:50; then 0>p2`1 by A54; hence contradiction by A54,A56,AXIOMS:22; end; then -p2`2>0 by A29,REAL_1:66; then (-p2`1)/(-p2`2)>=p2`2/(-p2`2) by A55,REAL_1:73; then (-p2`1)/(-p2`2)>= -1 by A29,XCMPLX_1:199; hence -1<=p2`1/p2`2 by XCMPLX_1:192; end; A57:now per cases by A16; case A58:p1`2<=p1`1 & -p1`1<=p1`2; now assume A59:p1`1<0; then 0<p1`2 by A58,REAL_1:66; hence contradiction by A58,A59,AXIOMS:22; end; then (-p1`1)/p1`1<=p1`2/p1`1 by A31,A58,REAL_1:73; hence -1<=p1`2/p1`1 by A31,XCMPLX_1:198; case A60:p1`2>=p1`1 & p1`2<=-p1`1; then A61: -p1`2>=--p1`1 by REAL_1:50; now assume A62:p1`1>0; then -p1`1< -0 by REAL_1:50; then 0>p1`2 by A60; hence contradiction by A60,A62,AXIOMS:22; end; then -p1`1>0 by A31,REAL_1:66; then (-p1`2)/(-p1`1)>=p1`1/(-p1`1) by A61,REAL_1:73; then (-p1`2)/(-p1`1)>= -1 by A31,XCMPLX_1:199; hence -1<=p1`2/p1`1 by XCMPLX_1:192; end; now per cases; case A63:0<=p2`1/p2`2; then p1`2>0 & p1`1>=0 or p1`2<0 & p1`1<=0 by A38,A39,REAL_2:134; then A64:p1`2/p1`1>=0 by REAL_2:125; now assume p1`2/p1`1<>1; then p1`2/p1`1<1 by A46,REAL_1:def 5; hence contradiction by A35,A41,A63,A64,REAL_2:139; end; then p1`2=(1)*p1`1 by A31,XCMPLX_1:88; then (p2`1/p2`2)*p2`2=(1)*p2`2 by A31,A39,XCMPLX_1:60.=p2`2; then p2`1=p2`2 by A29,XCMPLX_1:88; hence contradiction by A26,A40; case A65:0>p2`1/p2`2; then p1`2<0 & p1`1>0 or p1`2>0 & p1`1<0 by A38,A39,REAL_2:135; then A66:p1`2/p1`1<0 by REAL_2:128; now assume p1`2/p1`1<>-1; then -1<p1`2/p1`1 by A57,REAL_1:def 5; hence contradiction by A35,A51,A65,A66,REAL_2:139; end; then p1`2=(-1)*p1`1 by A31,XCMPLX_1:88 .= -p1`1 by XCMPLX_1:180; then -p1`2 =p1`1; then p2`1/p2`2=-1 by A38,A39,XCMPLX_1:198; then p2`1=(-1)*p2`2 by A29,XCMPLX_1:88; then -p2`1=--p2`2 by XCMPLX_1:181 .=p2`2; hence contradiction by A26,A40; end; hence contradiction; end; hence x1=x2; case x1 in K1a; then consider p7 being Point of TOP-REAL 2 such that A67: p1=p7 & ( (p7`1<=p7`2 & -p7`2<=p7`1 or p7`1>=p7`2 & p7`1<=-p7`2) & p7<>0.REAL 2); A68:Out_In_Sq.p1=|[p1`1/p1`2/p1`2,1/p1`2]| by A67,Th24; now per cases by A12,A15,XBOOLE_0:def 2; case x2 in K1a; then consider p8 being Point of (TOP-REAL 2) such that A69: p2=p8 & ( (p8`1<=p8`2 & -p8`2<=p8`1 or p8`1>=p8`2 & p8`1<=-p8`2) & p8<>0.REAL 2); A70: |[p2`1/p2`2/p2`2,1/p2`2]| =|[p1`1/p1`2/p1`2,1/p1`2]| by A7,A68,A69,Th24; set qq=|[p2`1/p2`2/p2`2,1/p2`2]|; qq`2=1/p2`2 & qq`1=p2`1/p2`2/p2`2 by EUCLID:56; then A71:1/p1`2= 1/p2`2 & p1`1/p1`2/p1`2 = p2`1/p2`2/p2`2 by A70,EUCLID:56; A72:(1/p1`2)"=(p1`2)"" by XCMPLX_1:217 .=p1`2; A73:(1/p2`2)"=(p2`2)"" by XCMPLX_1:217 .=p2`2; A74:now assume A75:p1`2=0; then p1`1=0 by A67; hence contradiction by A67,A75,EUCLID:57,58; end; then p1`1/p1`2= p2`1/p1`2 by A71,A72,A73,XCMPLX_1:53; then A76:p1`1=p2`1 by A74,XCMPLX_1:53; p1=|[p1`1,p1`2]| by EUCLID:57; hence x1=x2 by A71,A72,A73,A76,EUCLID:57; case A77:x2 in K0a & not x2 in K1a; then consider p8 being Point of (TOP-REAL 2) such that A78: p2=p8 & ( (p8`2<=p8`1 & -p8`1<=p8`2 or p8`2>=p8`1 & p8`2<=-p8`1) & p8<>0.REAL 2); Out_In_Sq.p2=|[1/p2`1,p2`2/p2`1/p2`1]| by A78,Def1; then A79:1/p1`2=p2`2/p2`1/p2`1 & p1`1/p1`2/p1`2=1/p2`1 by A7,A68,SPPOL_2:1; A80:now assume A81:p2`1=0; then p2`2=0 by A78; hence contradiction by A78,A81,EUCLID:57,58; end; A82:now assume A83:p1`2=0; then p1`1=0 by A67; hence contradiction by A67,A83,EUCLID:57,58; end; A84:p2`2/p2`1=1/p1`2*p2`1 by A79,A80,XCMPLX_1:88 .= p2`1/p1`2 by XCMPLX_1:100; p1`1/p1`2=1/p2`1*p1`2 by A79,A82,XCMPLX_1:88 .= p1`2/p2`1 by XCMPLX_1:100; then A85:(p2`2/p2`1)* (p1`1/p1`2)=1 by A80,A82,A84,XCMPLX_1:113; then (p2`2/p2`1)* (p1`1/p1`2)*p1`2=p1`2; then (p2`2/p2`1)* ((p1`1/p1`2)*p1`2)=p1`2 by XCMPLX_1:4; then A86:(p2`2/p2`1)*p1`1=p1`2 by A82,XCMPLX_1:88; A87:p2`2<>0 & p1`1<>0 by A85; then A88:(p2`2/p2`1)=p1`2/p1`1 by A86,XCMPLX_1:90; consider p9 being Point of (TOP-REAL 2) such that A89: p2=p9 & ( (p9`2<=p9`1 & -p9`1<=p9`2 or p9`2>=p9`1 & p9`2<=-p9`1) & p9<>0.REAL 2) by A77; A90:now per cases by A89; case A91:p2`2<=p2`1 & -p2`1<=p2`2; now assume A92:p2`1<0; then 0<p2`2 by A91,REAL_1:66; hence contradiction by A91,A92,AXIOMS:22; end; then p2`2/p2`1<=p2`1/p2`1 by A80,A91,REAL_1:73; hence p2`2/p2`1<=1 by A80,XCMPLX_1:60; case A93:p2`2>=p2`1 & p2`2<=-p2`1; now assume A94:p2`1>0; then -p2`1< -0 by REAL_1:50; then 0>p2`2 by A93; hence contradiction by A93,A94,AXIOMS:22; end; then p2`2/p2`1<=p2`1/p2`1 by A80,A93,REAL_1:74; hence p2`2/p2`1<=1 by A80,XCMPLX_1:60; end; A95:now per cases by A67; case A96:p1`1<=p1`2 & -p1`2<=p1`1; now assume A97:p1`2<0; then 0<p1`1 by A96,REAL_1:66; hence contradiction by A96,A97,AXIOMS:22; end; then p1`1/p1`2<=p1`2/p1`2 by A82,A96,REAL_1:73; hence p1`1/p1`2<=1 by A82,XCMPLX_1:60; case A98:p1`1>=p1`2 & p1`1<=-p1`2; now assume A99:p1`2>0; then -p1`2< -0 by REAL_1:50; then 0>p1`1 by A98; hence contradiction by A98,A99,AXIOMS:22; end; then p1`1/p1`2<=p1`2/p1`2 by A82,A98,REAL_1:74; hence p1`1/p1`2<=1 by A82,XCMPLX_1:60; end; A100:now per cases by A89; case A101:p2`2<=p2`1 & -p2`1<=p2`2; now assume A102:p2`1<0; then 0<p2`2 by A101,REAL_1:66; hence contradiction by A101,A102,AXIOMS:22; end; then (-p2`1)/p2`1<=p2`2/p2`1 by A80,A101,REAL_1:73; hence -1<=p2`2/p2`1 by A80,XCMPLX_1:198; case A103:p2`2>=p2`1 & p2`2<=-p2`1; then A104: -p2`2>=--p2`1 by REAL_1:50; now assume A105:p2`1>0; then -p2`1< -0 by REAL_1:50; then 0>p2`2 by A103; hence contradiction by A103,A105,AXIOMS:22; end; then -p2`1>0 by A80,REAL_1:66; then (-p2`2)/(-p2`1)>=p2`1/(-p2`1) by A104,REAL_1:73; then (-p2`2)/(-p2`1)>= -1 by A80,XCMPLX_1:199; hence -1<=p2`2/p2`1 by XCMPLX_1:192; end; A106:now per cases by A67; case A107:p1`1<=p1`2 & -p1`2<=p1`1; now assume A108:p1`2<0; then 0<p1`1 by A107,REAL_1:66; hence contradiction by A107,A108,AXIOMS:22; end; then (-p1`2)/p1`2<=p1`1/p1`2 by A82,A107,REAL_1:73; hence -1<=p1`1/p1`2 by A82,XCMPLX_1:198; case A109:p1`1>=p1`2 & p1`1<=-p1`2; then A110: -p1`1>=--p1`2 by REAL_1:50; now assume A111:p1`2>0; then -p1`2< -0 by REAL_1:50; then 0>p1`1 by A109; hence contradiction by A109,A111,AXIOMS:22; end; then -p1`2>0 by A82,REAL_1:66; then (-p1`1)/(-p1`2)>=p1`2/(-p1`2) by A110,REAL_1:73; then (-p1`1)/(-p1`2)>= -1 by A82,XCMPLX_1:199; hence -1<=p1`1/p1`2 by XCMPLX_1:192; end; now per cases; case A112:0<=p2`2/p2`1; then p1`1>0 & p1`2>=0 or p1`1<0 & p1`2<=0 by A87,A88,REAL_2:134; then A113:p1`1/p1`2>=0 by REAL_2:125; now assume p1`1/p1`2<>1; then p1`1/p1`2<1 by A95,REAL_1:def 5; hence contradiction by A85,A90,A112,A113,REAL_2:139; end; then p1`1=(1)*p1`2 by A82,XCMPLX_1:88; then (p2`2/p2`1)*p2`1 =(1)*p2`1 by A82,A88,XCMPLX_1:60 .=p2`1; then p2`2=p2`1 by A80,XCMPLX_1:88; hence contradiction by A77,A89; case A114:0>p2`2/p2`1; then p1`1<0 & p1`2>0 or p1`1>0 & p1`2<0 by A87,A88,REAL_2:135; then A115:p1`1/p1`2<0 by REAL_2:128; now assume p1`1/p1`2<>-1; then -1<p1`1/p1`2 by A106,REAL_1:def 5; hence contradiction by A85,A100,A114,A115,REAL_2:139; end; then p1`1=(-1)*p1`2 by A82,XCMPLX_1:88 .= -p1`2 by XCMPLX_1:180; then -p1`1 =p1`2; then p2`2/p2`1=-1 by A87,A88,XCMPLX_1:198; then p2`2=(-1)*p2`1 by A80,XCMPLX_1:88; then -p2`2=--p2`1 by XCMPLX_1:181 .=p2`1; hence contradiction by A77,A89; end; hence contradiction; end; hence x1=x2; end; hence x1=x2; end; then A116:Out_In_Sq is one-to-one by FUNCT_1:def 8; A117: for t being Point of TOP-REAL 2 st t in K0 & t<>0.REAL 2 holds not Out_In_Sq.t in K0 \/ Kb proof let t be Point of TOP-REAL 2; assume A118: t in K0 & t<>0.REAL 2; then consider p3 being Point of TOP-REAL 2 such that A119: p3=t & (-1<p3`1 & p3`1<1 & -1<p3`2 & p3`2<1) by A1; now assume A120: Out_In_Sq.t in K0 \/ Kb; now per cases by A120,XBOOLE_0:def 2; case Out_In_Sq.t in K0; then consider p4 being Point of TOP-REAL 2 such that A121: p4=Out_In_Sq.t & (-1<p4`1 & p4`1<1 & -1<p4`2 & p4`2<1) by A1; now per cases; case A122:(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A118,Def1; then A123:p4`1=1/t`1 & p4`2=t`2/t`1/t`1 by A121,EUCLID:56; now per cases; case A124: t`1>=0; now per cases by A124; case A125:t`1>0; then 1/t`1*t`1<1 *t`1 by A121,A123,REAL_1:70; hence contradiction by A119,A125,XCMPLX_1:88; case A126:t`1=0; then t`2=0 by A122; hence contradiction by A118,A126,EUCLID:57,58; end; hence contradiction; case A127:t`1<0; then (-1)*t`1>1/t`1*t`1 by A121,A123,REAL_1:71; then (-1)*t`1>1 by A127,XCMPLX_1:88; then -t`1>1 by XCMPLX_1:180; then --t`1<=-1 by REAL_1:50; hence contradiction by A119; end; hence contradiction; case A128:not(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A118,Def1; then A129:p4`2=1/t`2 & p4`1=t`1/t`2/t`2 by A121,EUCLID:56; A130: t`2> -t`1 implies -t`2<--t`1 by REAL_1:50; A131: t`2< -t`1 implies -t`2>--t`1 by REAL_1:50; now per cases; case A132: t`2>=0; now per cases by A132; case A133:t`2>0; then 1/t`2*t`2<1 *t`2 by A121,A129,REAL_1:70; hence contradiction by A119,A133,XCMPLX_1:88; case t`2=0; hence contradiction by A128,A130,A131; end; hence contradiction; case A134:t`2<0; then (-1)*t`2>1/t`2*t`2 by A121,A129,REAL_1:71; then (-1)*t`2>1 by A134,XCMPLX_1:88; then -t`2>1 by XCMPLX_1:180; then --t`2<=-1 by REAL_1:50; hence contradiction by A119; end; hence contradiction; end; hence contradiction; case Out_In_Sq.t in Kb; then consider p4 being Point of TOP-REAL 2 such that A135: p4=Out_In_Sq.t & (-1=p4`1 & -1<=p4`2 & p4`2<=1 or p4`1=1 & -1<=p4`2 & p4`2<=1 or -1=p4`2 & -1<=p4`1 & p4`1<=1 or 1=p4`2 & -1<=p4`1 & p4`1<=1) by A1; now per cases; case A136:(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then A137: Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A118,Def1; then A138:p4`1=1/t`1 & p4`2=t`2/t`1/t`1 by A135,EUCLID:56; now per cases by A135; case -1=p4`1 & -1<=p4`2 & p4`2<=1; then 1 *(t`1)"=-1 by A138,XCMPLX_0:def 9; then A139:(t`1)*(t`1)" =-t`1 by XCMPLX_1:180; now per cases; case t`1<>0; then -t`1=1 by A139,XCMPLX_0:def 7; hence contradiction by A119; case A140:t`1=0; then t`2=0 by A136; hence contradiction by A118,A140,EUCLID:57,58; end; hence contradiction; case p4`1=1 & -1<=p4`2 & p4`2<=1; then 1 *(t`1)"=1 by A138,XCMPLX_0:def 9; then A141:(t`1)*(t`1)"=t`1; now per cases; case t`1<>0; hence contradiction by A119,A141,XCMPLX_0:def 7; case A142:t`1=0; then t`2=0 by A136; hence contradiction by A118,A142,EUCLID:57,58; end; hence contradiction; case A143: -1=p4`2 & -1<=p4`1 & p4`1<=1; reconsider K01=K0a as non empty Subset of (TOP-REAL 2)|D by A3,Th27; A144:rng (Out_In_Sq|K01) c= the carrier of ((TOP-REAL 2)|D)|K01 by Th25; A145:dom (Out_In_Sq|K01)=(dom Out_In_Sq) /\ K01 by FUNCT_1:68 .=D /\ K01 by A2,FUNCT_2:def 1 .=[#]((TOP-REAL 2)|D) /\ K01 by PRE_TOPC:def 10 .=(the carrier of ((TOP-REAL 2)|D)) /\ K01 by PRE_TOPC:12 .=K01 by XBOOLE_1:28; t in K01 by A118,A136; then (Out_In_Sq|K01).t in rng (Out_In_Sq|K01) by A145,FUNCT_1:12; then A146:(Out_In_Sq|K01).t in the carrier of ((TOP-REAL 2)|D)|K01 by A144; A147:the carrier of ((TOP-REAL 2)|D)|K01=[#](((TOP-REAL 2)|D)|K01) by PRE_TOPC:12 .=K01 by PRE_TOPC:def 10; t in K01 by A118,A136; then Out_In_Sq.t in K0a by A146,A147,FUNCT_1:72; then consider p5 being Point of TOP-REAL 2 such that A148: p5=p4 & (p5`2<=p5`1 & -p5`1<=p5`2 or p5`2>=p5`1 & p5`2<=-p5`1) & p5<>0.REAL 2 by A135; now per cases by A143,A148,REAL_1:50; case p4`1>=1; then A149:1/t`1=1 by A138,A143,AXIOMS:21; then t`2/t`1/t`1=(t`2/t`1)*1 by XCMPLX_1:100 .=t`2*1 by A149, XCMPLX_1:100 .=t`2; hence contradiction by A119,A135,A137,A143,EUCLID:56; case -1>=p4`1; then A150:1/t`1=-1 by A138,A143,AXIOMS:21; then t`2/t`1/t`1=(t`2/t`1)*(-1) by XCMPLX_1:100 .=-(t`2/t`1) by XCMPLX_1:180 .=-(t`2*(-1)) by A150,XCMPLX_1:100 .= --t`2 by XCMPLX_1:181 .=t`2; hence contradiction by A119,A135,A137,A143,EUCLID:56; end; hence contradiction; case A151:1=p4`2 & -1<=p4`1 & p4`1<=1; reconsider K01=K0a as non empty Subset of (TOP-REAL 2)|D by A3,Th27; A152:rng (Out_In_Sq|K01) c= the carrier of ((TOP-REAL 2)|D)|K01 by Th25; dom (Out_In_Sq|K01)=(dom Out_In_Sq) /\ K01 by FUNCT_1:68 .=D /\ K01 by A2,FUNCT_2:def 1 .=[#]((TOP-REAL 2)|D) /\ K01 by PRE_TOPC:def 10 .=(the carrier of ((TOP-REAL 2)|D)) /\ K01 by PRE_TOPC:12 .=K01 by XBOOLE_1:28; then t in dom (Out_In_Sq|K01) by A118,A136; then (Out_In_Sq|K01).t in rng (Out_In_Sq|K01) by FUNCT_1:12; then A153:(Out_In_Sq|K01).t in the carrier of ((TOP-REAL 2)|D)|K01 by A152; A154:the carrier of ((TOP-REAL 2)|D)|K01=[#](((TOP-REAL 2)|D)|K01) by PRE_TOPC:12 .=K01 by PRE_TOPC:def 10; t in K01 by A118,A136; then Out_In_Sq.t=(Out_In_Sq|K01).t by FUNCT_1:72; then consider p5 being Point of TOP-REAL 2 such that A155: p5=p4 & (p5`2<=p5`1 & -p5`1<=p5`2 or p5`2>=p5`1 & p5`2<=-p5`1) & p5<>0.REAL 2 by A135,A153,A154; now per cases by A151,A155,REAL_2:109; case p4`1>=1; then A156:1/t`1=1 by A138,A151,AXIOMS:21; then t`2/t`1/t`1=(t`2/t`1)*1 by XCMPLX_1:100 .=t`2*1 by A156, XCMPLX_1:100 .=t`2; hence contradiction by A119,A135,A137,A151,EUCLID:56; case -1>=p4`1; then A157:1/t`1=-1 by A138,A151,AXIOMS:21; then t`2/t`1/t`1=(t`2/t`1)*(-1) by XCMPLX_1:100 .=-(t`2/t`1) by XCMPLX_1:180 .=-(t`2*(-1)) by A157,XCMPLX_1:100 .= --t`2 by XCMPLX_1:181 .=t`2; hence contradiction by A119,A135,A137,A151,EUCLID:56; end; hence contradiction; end; hence contradiction; case A158:not(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then A159: Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A118,Def1; then A160:p4`2=1/t`2 & p4`1=t`1/t`2/t`2 by A135,EUCLID:56; A161: t`2> -t`1 implies -t`2<--t`1 by REAL_1:50; A162: t`2< -t`1 implies -t`2>--t`1 by REAL_1:50; now per cases by A135; case -1=p4`2 & -1<=p4`1 & p4`1<=1; then 1 *(t`2)"=-1 by A160,XCMPLX_0:def 9; then A163:(t`2)*(t`2)"=-t`2 by XCMPLX_1:180; now per cases; case t`2<>0; then -t`2=1 by A163,XCMPLX_0:def 7; hence contradiction by A119; case t`2=0; hence contradiction by A158,A161,A162; end; hence contradiction; case p4`2=1 & -1<=p4`1 & p4`1<=1; then 1 *(t`2)"=1 by A160,XCMPLX_0:def 9; then A164:(t`2)*(t`2)" =t`2; now per cases; case t`2<>0; hence contradiction by A119,A164,XCMPLX_0:def 7; case t`2=0; hence contradiction by A158,A161,A162; end; hence contradiction; case A165: -1=p4`1 & -1<=p4`2 & p4`2<=1; A166:(t`1<=t`2 & -t`2<=t`1 or t`1>=t`2 & t`1<=-t`2) by A158,Th23; reconsider K11=K1a as non empty Subset of (TOP-REAL 2)|D by A3,Th28; A167:rng (Out_In_Sq|K11) c= the carrier of ((TOP-REAL 2)|D)|K11 by Th26; A168:dom (Out_In_Sq|K11)=(dom Out_In_Sq) /\ K11 by FUNCT_1:68 .=D /\ K11 by A2,FUNCT_2:def 1 .=[#]((TOP-REAL 2)|D) /\ K11 by PRE_TOPC:def 10 .=(the carrier of ((TOP-REAL 2)|D)) /\ K11 by PRE_TOPC:12 .=K11 by XBOOLE_1:28; t in K11 by A118,A166; then (Out_In_Sq|K11).t in rng (Out_In_Sq|K11) by A168,FUNCT_1:12; then A169:(Out_In_Sq|K11).t in the carrier of ((TOP-REAL 2)|D)|K11 by A167; A170:the carrier of ((TOP-REAL 2)|D)|K11=[#](((TOP-REAL 2)|D)|K11) by PRE_TOPC:12 .=K11 by PRE_TOPC:def 10; t in K11 by A118,A166; then Out_In_Sq.t=(Out_In_Sq|K11).t by FUNCT_1:72; then consider p5 being Point of TOP-REAL 2 such that A171: p5=p4 & (p5`1<=p5`2 & -p5`2<=p5`1 or p5`1>=p5`2 & p5`1<=-p5`2) & p5<>0.REAL 2 by A135,A169,A170; now per cases by A165,A171,REAL_1:50; case p4`2>=1; then A172:1/t`2=1 by A160,A165,AXIOMS:21; then t`1/t`2/t`2=(t`1/t`2)*1 by XCMPLX_1:100 .=t`1*1 by A172, XCMPLX_1:100 .=t`1; hence contradiction by A119,A135,A159,A165,EUCLID:56; case -1>=p4`2; then A173:1/t`2=-1 by A160,A165,AXIOMS:21; then t`1/t`2/t`2=(t`1/t`2)*(-1) by XCMPLX_1:100 .=-(t`1/t`2) by XCMPLX_1:180 .=-(t`1*(-1)) by A173,XCMPLX_1:100 .= --t`1 by XCMPLX_1:181 .=t`1; hence contradiction by A119,A135,A159,A165,EUCLID:56; end; hence contradiction; case A174:1=p4`1 & -1<=p4`2 & p4`2<=1; A175:(t`1<=t`2 & -t`2<=t`1 or t`1>=t`2 & t`1<=-t`2) by A158,Th23; reconsider K11=K1a as non empty Subset of (TOP-REAL 2)|D by A3,Th28; A176:rng (Out_In_Sq|K11) c= the carrier of ((TOP-REAL 2)|D)|K11 by Th26; A177:dom (Out_In_Sq|K11)=(dom Out_In_Sq) /\ K11 by FUNCT_1:68 .=D /\ K11 by A2,FUNCT_2:def 1 .=[#]((TOP-REAL 2)|D) /\ K11 by PRE_TOPC:def 10 .=(the carrier of ((TOP-REAL 2)|D)) /\ K11 by PRE_TOPC:12 .=K11 by XBOOLE_1:28; t in K11 by A118,A175; then (Out_In_Sq|K11).t in rng (Out_In_Sq|K11) by A177,FUNCT_1:12; then A178:(Out_In_Sq|K11).t in the carrier of ((TOP-REAL 2)|D)|K11 by A176; A179:the carrier of ((TOP-REAL 2)|D)|K11=[#](((TOP-REAL 2)|D)|K11) by PRE_TOPC:12 .=K11 by PRE_TOPC:def 10; t in K11 by A118,A175; then Out_In_Sq.t in K1a by A178,A179,FUNCT_1:72; then consider p5 being Point of TOP-REAL 2 such that A180: p5=p4 & (p5`1<=p5`2 & -p5`2<=p5`1 or p5`1>=p5`2 & p5`1<=-p5`2) & p5<>0.REAL 2 by A135; now per cases by A174,A180,REAL_2:109; case p4`2>=1; then A181:1/t`2=1 by A160,A174,AXIOMS:21; then t`1/t`2/t`2=(t`1/t`2)*1 by XCMPLX_1:100 .=t`1*1 by A181, XCMPLX_1:100 .=t`1; hence contradiction by A119,A135,A159,A174,EUCLID:56; case -1>=p4`2; then A182:1/t`2=-1 by A160,A174,AXIOMS:21; then t`1/t`2/t`2=(t`1/t`2)*(-1) by XCMPLX_1:100 .=-(t`1/t`2) by XCMPLX_1:180 .=-(t`1*(-1)) by A182,XCMPLX_1:100 .= --t`1 by XCMPLX_1:180 .=t`1; hence contradiction by A119,A135,A159,A174,EUCLID:56; end; hence contradiction; end; hence contradiction; end; hence contradiction; end; hence contradiction; end; hence not Out_In_Sq.t in K0 \/ Kb; end; A183: for t being Point of TOP-REAL 2 st not t in K0 \/ Kb holds Out_In_Sq.t in K0 proof let t be Point of TOP-REAL 2; assume not t in K0 \/ Kb; then A184:not t in K0 & not t in Kb by XBOOLE_0:def 2; then A185: not t=0.REAL 2 by A1,Th11; then not t in {0.REAL 2} by TARSKI:def 1; then t in (the carrier of TOP-REAL 2)\{0.REAL 2} by XBOOLE_0:def 4; then A186:Out_In_Sq.t in (the carrier of TOP-REAL 2)\{0.REAL 2} by FUNCT_2:7; (the carrier of TOP-REAL 2)\{0.REAL 2} c= the carrier of TOP-REAL 2 by XBOOLE_1:36; then reconsider p4=Out_In_Sq.t as Point of TOP-REAL 2 by A186; now per cases; case A187:(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A185,Def1; then A188:p4`1=1/t`1 & p4`2=t`2/t`1/t`1 by EUCLID:56; now per cases; case A189:t`1>0; then (t`1)">0 by REAL_1:72; then A190:1/t`1>0 by XCMPLX_1:217; A191:t`1>-1 by A189,AXIOMS:22; now per cases; case A192:t`2>0; A193: -0>-t`1 by A189,REAL_1:50; A194:t`2>-1 by A192,AXIOMS:22; -1>=t`1 or t`1>=1 or -1>=t`2 or t`2>=1 by A1,A184; then A195:t`1>=1 by A187,A192,A193,A194,AXIOMS:22; not t`1=1 by A1,A184,A187,AXIOMS:22; then A196:t`1>1 by A195,REAL_1:def 5; then A197: t`1/t`1>1/t`1 by A189,REAL_1:73; A198:0<t`2/t`1 by A189,A192,REAL_2:127; -t`1< -0 by A189,REAL_1:50; then (-1)*t`1<0 by XCMPLX_1:180; then (-1)*t`1 < t`2/t`1 by A198,AXIOMS:22; then A199: (-1)*t`1/t`1< t`2/t`1/t`1 by A189,REAL_1:73; t`1<(t`1)^2 by A196,SQUARE_1:76; then (t`2)<(t`1)^2 by A187,A192,A193,AXIOMS:22; then t`2/t`1<(t`1)^2/t`1 by A189,REAL_1:73; then t`2/t`1<(t`1)*(t`1)/t`1 by SQUARE_1:def 3; then t`2/t`1<(t`1) by A189,XCMPLX_1:90; then t`2/t`1/t`1<(t`1)/t`1 by A189,REAL_1:73; hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1 by A189,A190,A197,A199,AXIOMS:22,XCMPLX_1:60,90; case A200:t`2<=0; then A201:t`2<1 by AXIOMS:22; A202: --t`1>=-t`2 by A187,A189,A200,REAL_1:50; A203: now assume t`1<1; then -1>=t`2 by A1,A184,A191,A201; then -t`1<=-1 by A187,A189,A200,AXIOMS:22; hence t`1>=1 by REAL_1:50; end; not t`1=1 by A1,A184,A187,AXIOMS:22; then A204:t`1>1 by A203,REAL_1:def 5; then A205: t`1/t`1>1/t`1 by A189,REAL_1:73; t`1<(t`1)^2 by A204,SQUARE_1:76; then (t`1)^2 >-t`2 by A202,AXIOMS:22; then (t`1)^2/t`1 >(-t`2)/t`1 by A189,REAL_1:73; then t`1*t`1/t`1 >(-t`2)/t`1 by SQUARE_1:def 3; then t`1> (-t`2)/t`1 by A189,XCMPLX_1:90; then t`1>-(t`2/t`1) by XCMPLX_1:188; then -t`1<--(t`2/t`1) by REAL_1:50; then (-1)*t`1 < t`2/t`1 by XCMPLX_1:180; then A206: (-1)*t`1/t`1< t`2/t`1/t`1 by A189,REAL_1:73; t`1<(t`1)^2 by A204,SQUARE_1:76; then (t`2)<(t`1)^2 by A189,A200,AXIOMS:22; then t`2/t`1<(t`1)^2/t`1 by A189,REAL_1:73; then t`2/t`1<(t`1)*(t`1)/t`1 by SQUARE_1:def 3; then t`2/t`1<(t`1) by A189,XCMPLX_1:90; then t`2/t`1/t`1<(t`1)/t`1 by A189,REAL_1:73; hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1 by A189,A190,A205,A206,AXIOMS:22,XCMPLX_1:60,90; end; hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1; case A207: t`1<=0; now per cases by A207; case A208:t`1=0; then t`2=0 by A187; hence contradiction by A1,A184,A208; case A209:t`1<0; then A210:1/t`1<0 by REAL_2:149; A211: -t`1>-0 by A209,REAL_1:50; A212:t`1<1 by A209,AXIOMS:22; now per cases; case A213:t`2>0; -1>=t`1 or t`1>=1 or -1>=t`2 or t`2>=1 by A1,A184; then t`1<=-1 or 1<=-t`1 by A187,A209,A213,AXIOMS:22; then A214: t`1<=-1 or -1>=--t`1 by REAL_1:50; not t`1=-1 by A1,A184,A187,AXIOMS:22; then A215:t`1<-1 by A214,REAL_1:def 5; then t`1/t`1>(-1)/t`1 by A209,REAL_1:74; then -(t`1/t`1)<-((-1)/t`1) by REAL_1:50; then A216: -(t`1/t`1)<1/t`1 by XCMPLX_1:191; A217: 0>t`2/t`1 by A209,A213,REAL_2:128; -t`1> -0 by A209,REAL_1:50; then (-1)*t`1>0 by XCMPLX_1:180; then (-1)*t`1 > t`2/t`1 by A217,AXIOMS:22; then A218: (-1)*t`1/t`1< t`2/t`1/t`1 by A209,REAL_1:74; -t`1<(t`1)^2 by A215,Th4; then (t`2)<(t`1)^2 by A187,A209,A213,AXIOMS:22; then t`2/t`1>(t`1)^2/t`1 by A209,REAL_1:74; then t`2/t`1>(t`1)*(t`1)/t`1 by SQUARE_1:def 3; then t`2/t`1>(t`1) by A209,XCMPLX_1:90; then t`2/t`1/t`1<(t`1)/t`1 by A209,REAL_1:74; hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1 by A209,A210,A216,A218,AXIOMS:22,XCMPLX_1:60,90; case A219:t`2<=0; then t`2<1 by AXIOMS:22; then -1>=t`1 or -1>=t`2 by A1,A184,A212; then A220:t`1<=-1 by A187,A211,A219,AXIOMS:22; not t`1=-1 by A1,A184,A187,AXIOMS:22; then A221:t`1< -1 by A220,REAL_1:def 5; then t`1/t`1> (-1)/t`1 by A209,REAL_1:74; then 1> (-1)/t`1 by A209,XCMPLX_1:60; then A222: -1<-(-1)/t`1 by REAL_1:50; A223:-t`1>=-t`2 by A187,A211,A219,REAL_1:50; -t`1<(t`1)^2 by A221,Th4; then (t`1)^2 >-t`2 by A223,AXIOMS:22; then (t`1)^2/t`1 <(-t`2)/t`1 by A209,REAL_1:74; then t`1*t`1/t`1 <(-t`2)/t`1 by SQUARE_1:def 3; then t`1< (-t`2)/t`1 by A209,XCMPLX_1:90; then t`1<-(t`2/t`1) by XCMPLX_1:188; then -t`1>--(t`2/t`1) by REAL_1:50; then (-1)*t`1 > t`2/t`1 by XCMPLX_1:180; then A224: (-1)*t`1/t`1< t`2/t`1/t`1 by A209,REAL_1:74; -t`1<(t`1)^2 by A221,Th4; then (t`2)<(t`1)^2 by A211,A219,AXIOMS:22; then t`2/t`1>(t`1)^2/t`1 by A209,REAL_1:74; then t`2/t`1>(t`1)*(t`1)/t`1 by SQUARE_1:def 3; then t`2/t`1>(t`1) by A209,XCMPLX_1:90; then t`2/t`1/t`1<(t`1)/t`1 by A209,REAL_1:74; hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1 by A209,A210,A222,A224,AXIOMS:22,XCMPLX_1:60,90,191; end; hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1; end; hence -1<1/t`1 & 1/t`1<1 & -1< t`2/t`1/t`1 & t`2/t`1/t`1<1; end; hence Out_In_Sq.t in K0 by A1,A188; case A225:not(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A185,Def1; then A226:p4`2=1/t`2 & p4`1=t`1/t`2/t`2 by EUCLID:56; A227:t`1<=t`2 & -t`2<=t`1 or t`1>=t`2 & t`1<=-t`2 by A225,Th23; now per cases; case A228:t`2>0; then (t`2)">0 by REAL_1:72; then A229:1/t`2>0 by XCMPLX_1:217; A230:t`2>-1 by A228,AXIOMS:22; A231:t`1<=t`2 or t`1<=-t`2 by A225,Th23; now per cases; case A232:t`1>0; A233: -0>-t`2 by A228,REAL_1:50; A234:t`1>-1 by A232,AXIOMS:22; -1>=t`2 or t`2>=1 or -1>=t`1 or t`1>=1 by A1,A184; then A235:t`2>=1 by A231,A232,A233,A234,AXIOMS:22; not t`2=1 by A1,A184,A225,A234,Th23; then A236:t`2>1 by A235,REAL_1:def 5; then A237: t`2/t`2>1/t`2 by A228,REAL_1:73; A238:0<t`1/t`2 by A228,A232,REAL_2:127; -t`2< -0 by A228,REAL_1:50; then (-1)*t`2<0 by XCMPLX_1:180; then (-1)*t`2 < t`1/t`2 by A238,AXIOMS:22; then A239: (-1)*t`2/t`2< t`1/t`2/t`2 by A228,REAL_1:73; t`2<(t`2)^2 by A236,SQUARE_1:76; then (t`1)<(t`2)^2 by A231,A232,A233,AXIOMS:22; then t`1/t`2<(t`2)^2/t`2 by A228,REAL_1:73; then t`1/t`2<(t`2)*(t`2)/t`2 by SQUARE_1:def 3; then t`1/t`2<(t`2) by A228,XCMPLX_1:90; then t`1/t`2/t`2<(t`2)/t`2 by A228,REAL_1:73; hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by A228,A229,A237,A239,AXIOMS:22,XCMPLX_1:60,90; case A240:t`1<=0; then A241:t`1<1 by AXIOMS:22; A242: now assume t`2<1; then -1>=t`1 by A1,A184,A230,A241; then -t`2<=-1 by A227,A228,A240,AXIOMS:22; hence t`2>=1 by REAL_1:50; end; not t`2=1 by A1,A184,A227,AXIOMS:22; then A243:t`2>1 by A242,REAL_1:def 5; then A244: t`2/t`2>1/t`2 by A228,REAL_1:73; t`2<(t`2)^2 by A243,SQUARE_1:76; then (t`2)^2 >-t`1 by A225,A228,A240,AXIOMS:22; then (t`2)^2/t`2 >(-t`1)/t`2 by A228,REAL_1:73; then t`2*t`2/t`2 >(-t`1)/t`2 by SQUARE_1:def 3; then t`2> (-t`1)/t`2 by A228,XCMPLX_1:90; then t`2>-(t`1/t`2) by XCMPLX_1:188; then -t`2<--(t`1/t`2) by REAL_1:50; then (-1)*t`2 < t`1/t`2 by XCMPLX_1:180; then A245: (-1)*t`2/t`2< t`1/t`2/t`2 by A228,REAL_1:73; t`2<(t`2)^2 by A243,SQUARE_1:76; then (t`1)<(t`2)^2 by A228,A240,AXIOMS:22; then t`1/t`2<(t`2)^2/t`2 by A228,REAL_1:73; then t`1/t`2<(t`2)*(t`2)/t`2 by SQUARE_1:def 3; then t`1/t`2<(t`2) by A228,XCMPLX_1:90; then t`1/t`2/t`2<(t`2)/t`2 by A228,REAL_1:73; hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by A228,A229,A244,A245,AXIOMS:22,XCMPLX_1:60,90; end; hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1; case A246: t`2<=0; now per cases by A246; case A247:t`2=0; then t`1=0 by A227; hence contradiction by A1,A184,A247; case A248:t`2<0; then A249:1/t`2<0 by REAL_2:149; A250: -t`2>-0 by A248,REAL_1:50; A251:t`1<=t`2 or t`1<=-t`2 by A225,Th23; now per cases; case A252:t`1>0; -1>=t`2 or t`2>=1 or -1>=t`1 or t`1>=1 by A1,A184; then t`2<=-1 or 1<=-t`2 by A227,A248,A252,AXIOMS:22; then A253: t`2<=-1 or -1>=--t`2 by REAL_1:50; not t`2=-1 by A1,A184,A227,AXIOMS:22; then A254:t`2<-1 by A253,REAL_1:def 5; then t`2/t`2>(-1)/t`2 by A248,REAL_1:74; then -(t`2/t`2)<-((-1)/t`2) by REAL_1:50; then A255: -(t`2/t`2)<1/t`2 by XCMPLX_1:191; A256: 0>t`1/t`2 by A248,A252,REAL_2:128; -t`2> -0 by A248,REAL_1:50; then (-1)*t`2>0 by XCMPLX_1:180; then (-1)*t`2 > t`1/t`2 by A256,AXIOMS:22; then A257: (-1)*t`2/t`2< t`1/t`2/t`2 by A248,REAL_1:74; -t`2<(t`2)^2 by A254,Th4; then (t`1)<(t`2)^2 by A248,A251,A252,AXIOMS:22; then t`1/t`2>(t`2)^2/t`2 by A248,REAL_1:74; then t`1/t`2>(t`2)*(t`2)/t`2 by SQUARE_1:def 3; then t`1/t`2>(t`2) by A248,XCMPLX_1:90; then t`1/t`2/t`2<(t`2)/t`2 by A248,REAL_1:74; hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by A248,A249,A255,A257,AXIOMS:22,XCMPLX_1:60,90; case A258:t`1<=0; A259:t`2<1 by A248,AXIOMS:22; t`1<1 by A258,AXIOMS:22; then A260: -1>=t`2 or -1>=t`1 by A1,A184,A259; not t`2=-1 by A1,A184,A227,AXIOMS:22; then A261:t`2< -1 by A225,A250,A258,A260,Th23,AXIOMS:22,REAL_1:def 5; then t`2/t`2> (-1)/t`2 by A248,REAL_1:74; then 1> (-1)/t`2 by A248,XCMPLX_1:60; then A262: -1<-(-1)/t`2 by REAL_1:50; A263:-t`2>=-t`1 by A225,A250,A258,Th23,REAL_1:50; -t`2<(t`2)^2 by A261,Th4; then (t`2)^2 >-t`1 by A263,AXIOMS:22; then (t`2)^2/t`2 <(-t`1)/t`2 by A248,REAL_1:74; then t`2*t`2/t`2 <(-t`1)/t`2 by SQUARE_1:def 3; then t`2< (-t`1)/t`2 by A248,XCMPLX_1:90; then t`2<-(t`1/t`2) by XCMPLX_1:188; then -t`2>--(t`1/t`2) by REAL_1:50; then (-1)*t`2 > t`1/t`2 by XCMPLX_1:180; then A264: (-1)*t`2/t`2< t`1/t`2/t`2 by A248,REAL_1:74; -t`2<(t`2)^2 by A261,Th4; then (t`1)<(t`2)^2 by A250,A258,AXIOMS:22; then t`1/t`2>(t`2)^2/t`2 by A248,REAL_1:74; then t`1/t`2>(t`2)*(t`2)/t`2 by SQUARE_1:def 3; then t`1/t`2>(t`2) by A248,XCMPLX_1:90; then t`1/t`2/t`2<(t`2)/t`2 by A248,REAL_1:74; hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1 by A248,A249,A262,A264,AXIOMS:22,XCMPLX_1:60,90,191; end; hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1; end; hence -1<1/t`2 & 1/t`2<1 & -1< t`1/t`2/t`2 & t`1/t`2/t`2<1; end; hence Out_In_Sq.t in K0 by A1,A226; end; hence Out_In_Sq.t in K0; end; for s being Point of TOP-REAL 2 st s in Kb holds Out_In_Sq.s=s proof let t be Point of TOP-REAL 2; assume t in Kb; then consider p4 being Point of TOP-REAL 2 such that A265: p4=t & (-1=p4`1 & -1<=p4`2 & p4`2<=1 or p4`1=1 & -1<=p4`2 & p4`2<=1 or -1=p4`2 & -1<=p4`1 & p4`1<=1 or 1=p4`2 & -1<=p4`1 & p4`1<=1) by A1; A266:not t=0.REAL 2 by A265,EUCLID:56,58; A267:t<>0.REAL 2 by A265,EUCLID:56,58; now per cases; case A268:(t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then A269:Out_In_Sq.t=|[1/t`1,t`2/t`1/t`1]| by A266,Def1; A270: 1<=t`1 & t`1>=-1 or 1>=t`1 & -1>=--t`1 by A265,A268,REAL_1:50; now per cases by A265,A270,AXIOMS:21; case t`1=1; hence Out_In_Sq.t=t by A269,EUCLID:57; case A271:t`1=-1; then t`2/t`1/t`1 =(-t`2)/(-1) by XCMPLX_1:194 .=t`2 by XCMPLX_1:195; hence Out_In_Sq.t=t by A269,A271,EUCLID:57; end; hence Out_In_Sq.t=t; case A272:not (t`2<=t`1 & -t`1<=t`2 or t`2>=t`1 & t`2<=-t`1); then A273:Out_In_Sq.t=|[t`1/t`2/t`2,1/t`2]| by A267,Def1; now per cases by A265,A272; case t`2=1; hence Out_In_Sq.t=t by A273,EUCLID:57; case A274:t`2=-1; then t`1/t`2/t`2 =(-t`1)/(-1) by XCMPLX_1:194 .=t`1 by XCMPLX_1:195; hence Out_In_Sq.t=t by A273,A274,EUCLID:57; end; hence Out_In_Sq.t=t; end; hence Out_In_Sq.t=t; end; hence thesis by A4,A116,A117,A183; end; theorem Th52: for f,g being map of I[01],TOP-REAL 2, K0 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 & K0={p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1}& (f.O)`1=-1 & (f.I)`1=1 & -1<=(f.O)`2 & (f.O)`2<=1 & -1<=(f.I)`2 & (f.I)`2<=1 & (g.O)`2=-1 & (g.I)`2=1 & -1<=(g.O)`1 & (g.O)`1<=1 & -1<=(g.I)`1 & (g.I)`1<=1 & rng f misses K0 & rng g misses K0 holds rng f meets rng g proof let f,g be map of I[01],TOP-REAL 2, K0 be Subset of TOP-REAL 2, O,I be Point of I[01]; assume A1: O=0 & I=1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0={p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1}& (f.O)`1=-1 & (f.I)`1=1 & -1<=(f.O)`2 & (f.O)`2<=1 & -1<=(f.I)`2 & (f.I)`2<=1 & (g.O)`2=-1 & (g.I)`2=1 & -1<=(g.O)`1 & (g.O)`1<=1 & -1<=(g.I)`1 & (g.I)`1<=1 & rng f /\ K0={} & rng g /\ K0={}; defpred P[Point of TOP-REAL 2] means -1=$1`1 & -1<=$1`2 & $1`2<=1 or $1`1=1 & -1<=$1`2 & $1`2<=1 or -1=$1`2 & -1<=$1`1 & $1`1<=1 or 1=$1`2 & -1<=$1`1 & $1`1<=1; reconsider Kb={q: P[q]} as Subset of TOP-REAL 2 from TopSubset; reconsider B={0.REAL 2} as Subset of TOP-REAL 2; consider h being map of (TOP-REAL 2)|B`,(TOP-REAL 2)|B` such that A2:h is continuous & h is one-to-one & (for t being Point of TOP-REAL 2 st t in K0 & t<>0.REAL 2 holds not h.t in K0 \/ Kb) &(for r being Point of TOP-REAL 2 st not r in K0 \/ Kb holds h.r in K0) &(for s being Point of TOP-REAL 2 st s in Kb holds h.s=s) by A1,Th51; 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; A5:B`<>{} by Th19; rng f c= B` proof let x be set;assume A6:x in rng f; now assume x in B; then A7:x=0.REAL 2 by TARSKI:def 1; (0.REAL 2)`1=0 & (0.REAL 2)`2=0 by EUCLID:56,58; then 0.REAL 2 in K0 by A1; hence contradiction by A1,A6,A7,XBOOLE_0:def 3; end; then x in (the carrier of TOP-REAL 2)\ B by A6,XBOOLE_0:def 4; hence x in B` by SUBSET_1:def 5; end; then consider w being map of I[01],TOP-REAL 2 such that A8: w is continuous & w=h*f by A1,A2,A5,Th22; reconsider d1=h*f as map of I[01],TOP-REAL 2 by A8; A9:the carrier of (TOP-REAL 2)|B` =[#]((TOP-REAL 2)|B`) by PRE_TOPC:12 .=B` by PRE_TOPC:def 10; rng f c=(the carrier of (TOP-REAL 2))\B proof let e be set;assume A10:e in rng f; now assume e in B; then A11:e=0.REAL 2 by TARSKI:def 1; 0.REAL 2 in {p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} by Th11; hence contradiction by A1,A10,A11,XBOOLE_0:def 3; end; hence thesis by A10,XBOOLE_0:def 4; end; then A12:rng f c= the carrier of (TOP-REAL 2)|B` by A9,SUBSET_1:def 5; A13:d1 is one-to-one by A1,A2,FUNCT_1:46; rng g c= B` proof let x be set;assume A14:x in rng g; now assume x in B; then A15:x=0.REAL 2 by TARSKI:def 1; 0.REAL 2 in K0 by A1,Th11; hence contradiction by A1,A14,A15,XBOOLE_0:def 3; end; then x in (the carrier of TOP-REAL 2)\ B by A14,XBOOLE_0:def 4; hence x in B` by SUBSET_1:def 5; end; then consider w2 being map of I[01],TOP-REAL 2 such that A16: w2 is continuous & w2=h*g by A1,A2,A5,Th22; reconsider d2=h*g as map of I[01],TOP-REAL 2 by A16; rng g c=(the carrier of (TOP-REAL 2))\B proof let e be set;assume A17:e in rng g; now assume e in B; then A18:e=0.REAL 2 by TARSKI:def 1; 0.REAL 2 in {p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} by Th11; hence contradiction by A1,A17,A18,XBOOLE_0:def 3; end; hence thesis by A17,XBOOLE_0:def 4; end; then A19:rng g c= the carrier of (TOP-REAL 2)|B` by A9,SUBSET_1:def 5; A20:d2 is one-to-one by A1,A2,FUNCT_1:46; f.O in Kb by A1; then A21: h.(f.O)=f.O by A2; f.I in Kb by A1; then A22: h.(f.I)=f.I by A2; g.O in Kb by A1; then A23: h.(g.O)=g.O by A2; g.I in Kb by A1; then h.(g.I)=g.I by A2; then A24:(d1.O)`1=-1 & (d1.I)`1=1 & (d2.O)`2=-1 & (d2.I)`2=1 by A1,A3,A4,A21,A22,A23,FUNCT_1:23; for r being Point of I[01] holds -1<=(d1.r)`1 & (d1.r)`1<=1 & -1<=(d2.r)`1 & (d2.r)`1<=1 & -1<=(d1.r)`2 & (d1.r)`2<=1 & -1<=(d2.r)`2 & (d2.r)`2<=1 proof let r be Point of I[01]; f.r in rng f by A3,FUNCT_1:12; then A25:not f.r in K0 by A1,XBOOLE_0:def 3; A26:not f.r in Kb implies d1.r in K0 \/ Kb proof assume not f.r in Kb; then not f.r in K0 \/ Kb by A25,XBOOLE_0:def 2; then A27:h.(f.r) in K0 by A2; d1.r=h.(f.r) by A3,FUNCT_1:23; hence d1.r in K0 \/ Kb by A27,XBOOLE_0:def 2; end; A28: f.r in Kb implies d1.r in K0 \/ Kb proof assume A29:f.r in Kb; then A30:h.(f.r)=f.r by A2; d1.r=h.(f.r) by A3,FUNCT_1:23; hence d1.r in K0 \/ Kb by A29,A30,XBOOLE_0:def 2; end; g.r in rng g by A4,FUNCT_1:12; then A31:not g.r in K0 by A1,XBOOLE_0:def 3; A32:not g.r in Kb implies d2.r in K0 \/ Kb proof assume not g.r in Kb; then not g.r in K0 \/ Kb by A31,XBOOLE_0:def 2; then A33:h.(g.r) in K0 by A2; d2.r=h.(g.r) by A4,FUNCT_1:23; hence d2.r in K0 \/ Kb by A33,XBOOLE_0:def 2; end; A34: g.r in Kb implies d2.r in K0 \/ Kb proof assume A35:g.r in Kb; then A36:h.(g.r)=g.r by A2; d2.r=h.(g.r) by A4,FUNCT_1:23; hence d2.r in K0 \/ Kb by A35,A36,XBOOLE_0:def 2; end; now per cases by A26,A28,A32,A34,XBOOLE_0:def 2; case A37:d1.r in K0 & d2.r in K0; then consider p being Point of TOP-REAL 2 such that A38: p=d1.r & ( -1<p`1 & p`1<1 & -1<p`2 & p`2<1) by A1; consider q being Point of TOP-REAL 2 such that A39: q=d2.r & ( -1<q`1 & q`1<1 & -1<q`2 & q`2<1) by A1,A37; thus -1<=(d1.r)`1 & (d1.r)`1<=1 & -1<=(d2.r)`1 & (d2.r)`1<=1 & -1<=(d1.r)`2 & (d1.r)`2<=1 & -1<=(d2.r)`2 & (d2.r)`2<=1 by A38,A39; case A40:d1.r in K0 & d2.r in Kb; then consider p being Point of TOP-REAL 2 such that A41: p=d1.r & ( -1<p`1 & p`1<1 & -1<p`2 & p`2<1) by A1; consider q being Point of TOP-REAL 2 such that A42: q=d2.r & ( -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1) by A40; thus -1<=(d1.r)`1 & (d1.r)`1<=1 & -1<=(d2.r)`1 & (d2.r)`1<=1 & -1<=(d1.r)`2 & (d1.r)`2<=1 & -1<=(d2.r)`2 & (d2.r)`2<=1 by A41,A42; case A43:d1.r in Kb & d2.r in K0; then consider p being Point of TOP-REAL 2 such that A44: p=d2.r & ( -1<p`1 & p`1<1 & -1<p`2 & p`2<1) by A1; consider q being Point of TOP-REAL 2 such that A45: q=d1.r & ( -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1) by A43; thus -1<=(d1.r)`1 & (d1.r)`1<=1 & -1<=(d2.r)`1 & (d2.r)`1<=1 & -1<=(d1.r)`2 & (d1.r)`2<=1 & -1<=(d2.r)`2 & (d2.r)`2<=1 by A44,A45; case A46:d1.r in Kb & d2.r in Kb; then consider p being Point of TOP-REAL 2 such that A47: p=d2.r & ( -1=p`1 & -1<=p`2 & p`2<=1 or p`1=1 & -1<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1<=p`1 & p`1<=1); consider q being Point of TOP-REAL 2 such that A48: q=d1.r & ( -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1) by A46; thus -1<=(d1.r)`1 & (d1.r)`1<=1 & -1<=(d2.r)`1 & (d2.r)`1<=1 & -1<=(d1.r)`2 & (d1.r)`2<=1 & -1<=(d2.r)`2 & (d2.r)`2<=1 by A47,A48; end; hence -1<=(d1.r)`1 & (d1.r)`1<=1 & -1<=(d2.r)`1 & (d2.r)`1<=1 & -1<=(d1.r)`2 & (d1.r)`2<=1 & -1<=(d2.r)`2 & (d2.r)`2<=1; end; then rng d1 meets rng d2 by A1,A8,A13,A16,A20,A24,JGRAPH_1:65; then A49:rng d1 /\ rng d2<>{} by XBOOLE_0:def 7; consider s being Element of rng d1 /\ rng d2; A50:s in rng d1 & s in rng d2 by A49,XBOOLE_0:def 3; then consider t1 being set such that A51:t1 in dom d1 & s=d1.t1 by FUNCT_1:def 5; consider t2 being set such that A52:t2 in dom d2 & s=d2.t2 by A50,FUNCT_1:def 5; reconsider W=B` as non empty Subset of TOP-REAL 2 by Th19; A53: the carrier of (TOP-REAL 2)|W <>{}; h.(f.t1)=d1.t1 by A51,FUNCT_1:22; then A54:h.(f.t1)=h.(g.t2) by A51,A52,FUNCT_1:22; A55:dom h=the carrier of (TOP-REAL 2)|B` by A53,FUNCT_2:def 1; A56:f.t1 in rng f by A3,A51,FUNCT_1:12; dom g =the carrier of I[01] by FUNCT_2:def 1; then A57:g.t2 in rng g by A52,FUNCT_1:12; then f.t1=g.t2 by A2,A12,A19,A54,A55,A56,FUNCT_1:def 8; then rng f /\ rng g <> {} by A56,A57,XBOOLE_0:def 3; hence thesis by XBOOLE_0:def 7; end; theorem Th53:for A,B,C,D being real number, f being map of TOP-REAL 2,TOP-REAL 2 st (for t being Point of TOP-REAL 2 holds f.t=|[A*(t`1)+B,C*(t`2)+D]|) holds f is continuous proof let A,B,C,D be real number,f be map of TOP-REAL 2,TOP-REAL 2; assume A1: (for t being Point of TOP-REAL 2 holds f.t=|[A*(t`1)+B,C*(t`2)+D]|); A2:(TOP-REAL 2)| [#](TOP-REAL 2)=TOP-REAL 2 by TSEP_1:3; proj1*f is Function of the carrier of TOP-REAL 2,the carrier of R^1 by TOPMETR:24; then reconsider f1=proj1*f as map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 by A2; proj2*f is Function of the carrier of TOP-REAL 2,the carrier of R^1 by TOPMETR:24; then reconsider f2=proj2*f as map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 by A2; reconsider f0=f as map of (TOP-REAL 2)| [#](TOP-REAL 2), (TOP-REAL 2)| [#](TOP-REAL 2) by A2; set K0=[#](TOP-REAL 2); reconsider h11=proj1 as map of TOP-REAL 2,R^1 by TOPMETR:24; reconsider h1=h11 as map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 by A2; h11 is continuous by TOPREAL6:83; then consider g1 being map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that A5: (for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being real number st h1.p=r1 holds g1.p=A*r1) & g1 is continuous by A2,Th33; consider g11 being map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that A6: (for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being real number st g1.p=r1 holds g11.p=r1+B) & g11 is continuous by A5,Th34; dom f1=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then A7: dom f1=dom g11 by A2,FUNCT_2:def 1; for x being set st x in dom f1 holds f1.x=g11.x proof let x be set;assume A8: x in dom f1; then A9: f1.x=proj1.(f.x) by FUNCT_1:22; reconsider p=x as Point of TOP-REAL 2 by A8,FUNCT_2:def 1; A10:f1.x=proj1.(|[A*(p`1)+B,C*(p`2)+D]|) by A1,A9 .=A*(p`1)+B by JORDAN1A:20 .=A*(proj1.p)+B by PSCOMP_1:def 28; A*(proj1.p)=g1.p by A2,A5; hence f1.x=g11.x by A2,A6,A10; end; then A11:f1 is continuous by A6,A7,FUNCT_1:9; reconsider h11=proj2 as map of TOP-REAL 2,R^1 by TOPMETR:24; reconsider h1=h11 as map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 by A2; h1 is continuous by A2,TOPREAL6:83; then consider g1 being map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that A12:(for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being real number st h1.p=r1 holds g1.p=C*r1) & g1 is continuous by Th33; consider g11 being map of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that A13:(for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being real number st g1.p=r1 holds g11.p=r1+D) & g11 is continuous by A12,Th34; dom f2=the carrier of TOP-REAL 2 by FUNCT_2:def 1; then A14: dom f2=dom g11 by A2,FUNCT_2:def 1; for x being set st x in dom f2 holds f2.x=g11.x proof let x be set;assume A15: x in dom f2; then A16: f2.x=proj2.(f.x) by FUNCT_1:22; reconsider p=x as Point of TOP-REAL 2 by A15,FUNCT_2:def 1; A17:f2.x=proj2.(|[A*(p`1)+B,C*(p`2)+D]|) by A1,A16 .=C*(p`2)+D by JORDAN1A:20 .=C*(proj2.p)+D by PSCOMP_1:def 29; C*(proj2.p)=g1.p by A2,A12; hence f2.x=g11.x by A2,A13,A17; end; then A18:f2 is continuous by A13,A14,FUNCT_1:9; for x,y,r,s being real number st |[x,y]| in K0 & r=f1.(|[x,y]|) & s=f2.(|[x,y]|) holds f0. |[x,y]|=|[r,s]| proof let x,y,r,s be real number;assume A19: |[x,y]| in K0 & r=f1.(|[x,y]|) & s=f2.(|[x,y]|); A20: dom f =the carrier of TOP-REAL 2 by FUNCT_2:def 1; A21:f. |[x,y]| is Point of TOP-REAL 2; A22: proj1.(f0. |[x,y]|) =r by A19,A20,FUNCT_1:23; proj2.(f0. |[x,y]|) =s by A19,A20,FUNCT_1:23; hence f0. |[x,y]|=|[r,s]| by A21,A22,Th18; end; hence f is continuous by A2,A11,A18,Th45; end; definition let A,B,C,D be real number; func AffineMap(A,B,C,D) -> map of TOP-REAL 2,TOP-REAL 2 means :Def2: for t being Point of TOP-REAL 2 holds it.t=|[A*(t`1)+B,C*(t`2)+D]|; existence proof defpred P[set,set] means for t being Point of TOP-REAL 2 st t=$1 holds $2=|[A*(t`1)+B,C*(t`2)+D]|; A1: for x,y1,y2 being set st x in the carrier of TOP-REAL 2 & P[x,y1] & P[x,y2] holds y1 = y2 proof let x,y1,y2 be set; assume A2: x in the carrier of TOP-REAL 2 & P[x,y1] & P[x,y2]; then reconsider t=x as Point of TOP-REAL 2; y1=|[A*(t`1)+B,C*(t`2)+D]| by A2; hence y1 = y2 by A2; end; A3: for x being set st x in the carrier of TOP-REAL 2 ex y being set st P[x,y] proof let x be set;assume x in the carrier of TOP-REAL 2; then reconsider t2=x as Point of TOP-REAL 2; reconsider y2=|[A*(t2`1)+B,C*(t2`2)+D]| as set; (for t being Point of TOP-REAL 2 st t=x holds y2 =|[A*(t`1)+B,C*(t`2)+D]|); hence ex y being set st P[x,y]; end; ex ff being Function st dom ff=(the carrier of TOP-REAL 2) & for x being set st x in (the carrier of TOP-REAL 2) holds P[x,ff.x] from FuncEx(A1,A3); then consider ff being Function such that A4: dom ff=the carrier of TOP-REAL 2 & for x being set st x in the carrier of TOP-REAL 2 holds (for t being Point of TOP-REAL 2 st t=x holds ff.x=|[A*(t`1)+B,C*(t`2)+D]|); for x being set st x in the carrier of TOP-REAL 2 holds ff.x in the carrier of TOP-REAL 2 proof let x be set;assume x in the carrier of TOP-REAL 2; then reconsider t=x as Point of TOP-REAL 2; ff.t=|[A*(t`1)+B,C*(t`2)+D]| by A4; hence ff.x in the carrier of TOP-REAL 2; end; then ff is Function of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 by A4,FUNCT_2:5; then reconsider ff as map of TOP-REAL 2,TOP-REAL 2; take ff; thus thesis by A4; end; uniqueness proof let m1,m2 be map of TOP-REAL 2,TOP-REAL 2 such that A5: for t being Point of TOP-REAL 2 holds m1.t=|[A*(t`1)+B,C*(t`2)+D]| and A6: for t being Point of TOP-REAL 2 holds m2.t=|[A*(t`1)+B,C*(t`2)+D]|; for x being Point of TOP-REAL 2 holds m1.x = m2.x proof let t be Point of TOP-REAL 2; thus m1.t = |[A*(t`1)+B,C*(t`2)+D]| by A5 .= m2.t by A6; end; hence m1 = m2 by FUNCT_2:113; end; end; definition let a,b,c,d be real number; cluster AffineMap(a,b,c,d) -> continuous; coherence proof for t being Point of TOP-REAL 2 holds AffineMap(a,b,c,d).t=|[a*(t`1)+b,c*(t`2)+d]| by Def2; hence thesis by Th53; end; end; theorem Th54: for A,B,C,D being real number st A>0 & C>0 holds AffineMap(A,B,C,D) is one-to-one proof let A,B,C,D be real number such that A1:A>0 and A2:C>0; set ff = AffineMap(A,B,C,D); for x1,x2 being set st x1 in dom ff & x2 in dom ff & ff.x1=ff.x2 holds x1=x2 proof let x1,x2 be set;assume A3: x1 in dom ff & x2 in dom ff & ff.x1=ff.x2; then reconsider p1=x1 as Point of TOP-REAL 2; reconsider p2=x2 as Point of TOP-REAL 2 by A3; A4: ff.x1= |[A*(p1`1)+B,C*(p1`2)+D]| by Def2; ff.x2= |[A*(p2`1)+B,C*(p2`2)+D]| by Def2; then A*(p1`1)+B=A*(p2`1)+B & C*(p1`2)+D=C*(p2`2)+D by A3,A4,SPPOL_2:1; then A*(p1`1)=A*(p2`1)+B-B & C*(p1`2)+D-D=C*(p2`2)+D-D by XCMPLX_1:26; then A*(p1`1)=A*(p2`1) & C*(p1`2)=C*(p2`2)+D-D by XCMPLX_1:26; then (p1`1)=A*(p2`1)/A & C*(p1`2)/C=C*(p2`2)/C by A1,XCMPLX_1:26,90; then (p1`1)=(p2`1) & (p1`2)=C*(p2`2)/C by A1,A2,XCMPLX_1:90; then (p1`1)=(p2`1) & (p1`2)=(p2`2) by A2,XCMPLX_1:90; hence x1=x2 by TOPREAL3:11; end; hence ff is one-to-one by FUNCT_1:def 8; end; theorem for f,g being map of I[01],TOP-REAL 2,a,b,c,d being real number, O,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one & (f.O)`1=a & (f.I)`1=b & c <=(f.O)`2 & (f.O)`2<=d & c <=(f.I)`2 & (f.I)`2<=d & (g.O)`2=c & (g.I)`2=d & a<=(g.O)`1 & (g.O)`1<=b & a<=(g.I)`1 & (g.I)`1<=b & a < b & c < d & not (ex r being Point of I[01] st a<(f.r)`1 & (f.r)`1<b & c <(f.r)`2 & (f.r)`2<d)& not (ex r being Point of I[01] st a<(g.r)`1 & (g.r)`1<b & c <(g.r)`2 & (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 number, 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 & (f.O)`1=a & (f.I)`1=b & c <=(f.O)`2 & (f.O)`2<=d & c <=(f.I)`2 & (f.I)`2<=d & (g.O)`2=c & (g.I)`2=d & a<=(g.O)`1 & (g.O)`1<=b & a<=(g.I)`1 & (g.I)`1<=b & a < b & c < d & not (ex r being Point of I[01] st a<(f.r)`1 & (f.r)`1<b & c <(f.r)`2 & (f.r)`2<d)& not (ex r being Point of I[01] st a<(g.r)`1 & (g.r)`1<b & c <(g.r)`2 & (g.r)`2<d); then A2:b-a>0 by SQUARE_1:11; A3:d-c>0 by A1,SQUARE_1:11; set A=2/(b-a),B=1-2*b/(b-a),C=2/(d-c),D=1-2*d/(d-c); A4:A>0 by A2,REAL_2:127; A5:C>0 by A3,REAL_2:127; set ff =AffineMap(A,B,C,D); A6:dom ff=the carrier of TOP-REAL 2 by FUNCT_2:def 1; A7:dom f=the carrier of I[01] by FUNCT_2:def 1; A8:dom g=the carrier of I[01] by FUNCT_2:def 1; A9: ff is one-to-one by A4,A5,Th54; reconsider f2=ff*f,g2=ff*g as map of I[01],TOP-REAL 2; A10:f2 is continuous by A1,TOPS_2:58; A11:g2 is continuous by A1,TOPS_2:58; A12:f2 is one-to-one by A1,A9,FUNCT_1:46; A13:g2 is one-to-one by A1,A9,FUNCT_1:46; defpred P[Point of TOP-REAL 2] means -1<$1`1 & $1`1<1 & -1<$1`2 & $1`2<1; reconsider K0={p: P[p]} as Subset of TOP-REAL 2 from TopSubset; A14:f2.O=ff.(f.O) by A7,FUNCT_1:23 .=|[A*a+B,C*((f.O)`2)+D]| by A1,Def2; A15:f2.I=ff.(f.I) by A7,FUNCT_1:23 .=|[A*b+B,C*((f.I)`2)+D]| by A1,Def2; A16:g2.O=ff.(g.O) by A8,FUNCT_1:23 .=|[A*((g.O)`1)+B,C*c+D]| by A1,Def2; A17:g2.I=ff.(g.I) by A8,FUNCT_1:23 .=|[A*((g.I)`1)+B,C*d+D]| by A1,Def2; A18:(f2.O)`1= -1 proof thus (f2.O)`1=A*a+B by A14,EUCLID:56 .= a*2/(b-a)+(1-2*b/(b-a)) by XCMPLX_1:75 .= a*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A2,XCMPLX_1:60 .= a*2/(b-a)+((b-a)-2*b)/(b-a) by XCMPLX_1:121 .= (a*2+((b-a)-2*b))/(b-a) by XCMPLX_1:63 .= (2*a+(b-a)-2*b)/(b-a) by XCMPLX_1:29 .= (2*a-2*b+(b-a))/(b-a) by XCMPLX_1:29 .= (2*(a-b)+(b-a))/(b-a) by XCMPLX_1:40 .= (2*(a-b)-(a-b))/(b-a) by XCMPLX_1:38 .= ((a-b)+(a-b)-(a-b))/(b-a) by XCMPLX_1:11 .= (a-b)/(b-a) by XCMPLX_1:26 .= (-(b-a))/(b-a) by XCMPLX_1:143 .= -((b-a)/(b-a)) by XCMPLX_1:188 .= -1 by A2,XCMPLX_1:60; end; A19:(f2.I)`1=A*b+B by A15,EUCLID:56 .= b*2/(b-a)+(1-2*b/(b-a)) by XCMPLX_1:75 .= b*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A2,XCMPLX_1:60 .= b*2/(b-a)+((b-a)-2*b)/(b-a) by XCMPLX_1:121 .= (b*2+((b-a)-2*b))/(b-a) by XCMPLX_1:63 .= (2*b+(b-a)-2*b)/(b-a) by XCMPLX_1:29 .= (2*b-2*b+(b-a))/(b-a) by XCMPLX_1:29 .= (0+(b-a))/(b-a) by XCMPLX_1:14 .= 1 by A2,XCMPLX_1:60; A20:(g2.O)`2=2/(d-c)*c+(1-2*d/(d-c)) by A16,EUCLID:56 .= c*2/(d-c)+(1-2*d/(d-c)) by XCMPLX_1:75 .= c*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A3,XCMPLX_1:60 .= c*2/(d-c)+((d-c)-2*d)/(d-c) by XCMPLX_1:121 .= (c*2+((d-c)-2*d))/(d-c) by XCMPLX_1:63 .= (2*c+(d-c)-2*d)/(d-c) by XCMPLX_1:29 .= (2*c-2*d+(d-c))/(d-c) by XCMPLX_1:29 .= (2*(c-d)+(d-c))/(d-c) by XCMPLX_1:40 .= (2*(c-d)-(c-d))/(d-c) by XCMPLX_1:38 .= ((c-d)+(c-d)-(c-d))/(d-c) by XCMPLX_1:11 .= (c-d)/(d-c) by XCMPLX_1:26 .= (-(d-c))/(d-c) by XCMPLX_1:143 .= -((d-c)/(d-c)) by XCMPLX_1:188 .= -1 by A3,XCMPLX_1:60; A21: (g2.I)`2=C*d+D by A17,EUCLID:56 .= d*2/(d-c)+(1-2*d/(d-c)) by XCMPLX_1:75 .= d*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A3,XCMPLX_1:60 .= d*2/(d-c)+((d-c)-2*d)/(d-c) by XCMPLX_1:121 .= (d*2+((d-c)-2*d))/(d-c) by XCMPLX_1:63 .= (2*d+(d-c)-2*d)/(d-c) by XCMPLX_1:29 .= (2*d-2*d+(d-c))/(d-c) by XCMPLX_1:29 .= (0+(d-c))/(d-c) by XCMPLX_1:14 .= 1 by A3,XCMPLX_1:60; A22: -1<=(f2.O)`2 & (f2.O)`2<=1 & -1<=(f2.I)`2 & (f2.I)`2<=1 proof reconsider s0=(f.O)`2 as Real; A23:(f2.O)`2=((s0-d)+(s0-d)-(c-d))/(d-c) proof thus (f2.O)`2=C*s0+D by A14,EUCLID:56 .= s0 *2/(d-c)+(1-2*d/(d-c)) by XCMPLX_1:75 .= s0 *2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A3,XCMPLX_1:60 .= s0 *2/(d-c)+((d-c)-2*d)/(d-c) by XCMPLX_1:121 .= (s0 *2+((d-c)-2*d))/(d-c) by XCMPLX_1:63 .= (2*s0+(d-c)-2*d)/(d-c) by XCMPLX_1:29 .= (2*s0-2*d+(d-c))/(d-c) by XCMPLX_1:29 .= (2*(s0-d)+(d-c))/(d-c) by XCMPLX_1:40 .= (2*(s0-d)-(c-d))/(d-c) by XCMPLX_1:38 .= ((s0-d)+(s0-d)-(c-d))/(d-c) by XCMPLX_1:11; end; c-d<=s0-d by A1,REAL_1:49; then c-d+(c-d)<=(s0-d)+(s0-d) by REAL_1:55; then c-d+(c-d)-(c-d)<=(s0-d)+(s0-d)-(c-d) by REAL_1:49; then A24: c-d<=(s0-d)+(s0-d)-(c-d) by XCMPLX_1:26; A25: (c-d)/(d-c) = (-(d-c))/(d-c) by XCMPLX_1:143 .= -((d-c)/(d-c)) by XCMPLX_1:188 .= -1 by A3,XCMPLX_1:60; d-d>=s0-d by A1,REAL_1:49; then d-d+(d-d)>=(s0-d)+(s0-d) by REAL_1:55; then d-d+(d-d)-(c-d)>=(s0-d)+(s0-d)-(c-d) by REAL_1:49; then 0+(d-d)-(c-d)>=(s0-d)+(s0-d)-(c-d) by XCMPLX_1:14; then 0+0-(c-d)>=(s0-d)+(s0-d)-(c-d) by XCMPLX_1:14; then -(c-d)>=(s0-d)+(s0-d)-(c-d) by XCMPLX_1:150; then d-c >=(s0-d)+(s0-d)-(c-d) by XCMPLX_1:143; then A26:(d-c)/(d-c)>=((s0-d)+(s0-d)-(c-d))/(d-c) by A3,REAL_1:73; reconsider s1=(f.I)`2 as Real; A27:(f2.I)`2=((s1-d)+(s1-d)-(c-d))/(d-c) proof thus (f2.I)`2=C*s1+D by A15,EUCLID:56 .= s1*2/(d-c)+(1-2*d/(d-c)) by XCMPLX_1:75 .= s1*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A3,XCMPLX_1:60 .= s1*2/(d-c)+((d-c)-2*d)/(d-c) by XCMPLX_1:121 .= (s1*2+((d-c)-2*d))/(d-c) by XCMPLX_1:63 .= (2*s1+(d-c)-2*d)/(d-c) by XCMPLX_1:29 .= (2*s1-2*d+(d-c))/(d-c) by XCMPLX_1:29 .= (2*(s1-d)+(d-c))/(d-c) by XCMPLX_1:40 .= (2*(s1-d)-(c-d))/(d-c) by XCMPLX_1:38 .= ((s1-d)+(s1-d)-(c-d))/(d-c) by XCMPLX_1:11; end; c-d<=s1-d by A1,REAL_1:49; then c-d+(c-d)<=(s1-d)+(s1-d) by REAL_1:55; then c-d+(c-d)-(c-d)<=(s1-d)+(s1-d)-(c-d) by REAL_1:49; then A28: c-d<=(s1-d)+(s1-d)-(c-d) by XCMPLX_1:26; d-d>=s1-d by A1,REAL_1:49; then d-d+(d-d)>=(s1-d)+(s1-d) by REAL_1:55; then d-d+(d-d)-(c-d)>=(s1-d)+(s1-d)-(c-d) by REAL_1:49; then 0+(d-d)-(c-d)>=(s1-d)+(s1-d)-(c-d) by XCMPLX_1:14; then 0+0-(c-d)>=(s1-d)+(s1-d)-(c-d) by XCMPLX_1:14; then -(c-d)>=(s1-d)+(s1-d)-(c-d) by XCMPLX_1:150; then d-c >=(s1-d)+(s1-d)-(c-d) by XCMPLX_1:143; then (d-c)/(d-c)>=((s1-d)+(s1-d)-(c-d))/(d-c) by A3,REAL_1:73; hence thesis by A3,A23,A24,A25,A26,A27,A28,REAL_1:73,XCMPLX_1:60; end; A29: -1<=(g2.O)`1 & (g2.O)`1<=1 & -1<=(g2.I)`1 & (g2.I)`1<=1 proof reconsider s0=(g.O)`1 as Real; A30:(g2.O)`1=((s0-b)+(s0-b)-(a-b))/(b-a) proof thus (g2.O)`1=A*s0+B by A16,EUCLID:56 .= s0 *2/(b-a)+(1-2*b/(b-a)) by XCMPLX_1:75 .= s0 *2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A2,XCMPLX_1:60 .= s0 *2/(b-a)+((b-a)-2*b)/(b-a) by XCMPLX_1:121 .= (s0 *2+((b-a)-2*b))/(b-a) by XCMPLX_1:63 .= (2*s0+(b-a)-2*b)/(b-a) by XCMPLX_1:29 .= (2*s0-2*b+(b-a))/(b-a) by XCMPLX_1:29 .= (2*(s0-b)+(b-a))/(b-a) by XCMPLX_1:40 .= (2*(s0-b)-(a-b))/(b-a) by XCMPLX_1:38 .= ((s0-b)+(s0-b)-(a-b))/(b-a) by XCMPLX_1:11; end; a-b<=s0-b by A1,REAL_1:49; then a-b+(a-b)<=(s0-b)+(s0-b) by REAL_1:55; then a-b+(a-b)-(a-b)<=(s0-b)+(s0-b)-(a-b) by REAL_1:49; then A31: a-b<=(s0-b)+(s0-b)-(a-b) by XCMPLX_1:26; A32: (a-b)/(b-a) = (-(b-a))/(b-a) by XCMPLX_1:143 .= -((b-a)/(b-a)) by XCMPLX_1:188 .= -1 by A2,XCMPLX_1:60; b-b>=s0-b by A1,REAL_1:49; then b-b+(b-b)>=(s0-b)+(s0-b) by REAL_1:55; then b-b+(b-b)-(a-b)>=(s0-b)+(s0-b)-(a-b) by REAL_1:49; then 0+(b-b)-(a-b)>=(s0-b)+(s0-b)-(a-b) by XCMPLX_1:14; then 0+0-(a-b)>=(s0-b)+(s0-b)-(a-b) by XCMPLX_1:14; then -(a-b)>=(s0-b)+(s0-b)-(a-b) by XCMPLX_1:150; then b-a >=(s0-b)+(s0-b)-(a-b) by XCMPLX_1:143; then A33:(b-a)/(b-a)>=((s0-b)+(s0-b)-(a-b))/(b-a) by A2,REAL_1:73; reconsider s1=(g.I)`1 as Real; A34:(g2.I)`1=((s1-b)+(s1-b)-(a-b))/(b-a) proof thus (g2.I)`1=A*s1+B by A17,EUCLID:56 .= s1*2/(b-a)+(1-2*b/(b-a)) by XCMPLX_1:75 .= s1*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A2,XCMPLX_1:60 .= s1*2/(b-a)+((b-a)-2*b)/(b-a) by XCMPLX_1:121 .= (s1*2+((b-a)-2*b))/(b-a) by XCMPLX_1:63 .= (2*s1+(b-a)-2*b)/(b-a) by XCMPLX_1:29 .= (2*s1-2*b+(b-a))/(b-a) by XCMPLX_1:29 .= (2*(s1-b)+(b-a))/(b-a) by XCMPLX_1:40 .= (2*(s1-b)-(a-b))/(b-a) by XCMPLX_1:38 .= ((s1-b)+(s1-b)-(a-b))/(b-a) by XCMPLX_1:11; end; a-b<=s1-b by A1,REAL_1:49; then a-b+(a-b)<=(s1-b)+(s1-b) by REAL_1:55; then a-b+(a-b)-(a-b)<=(s1-b)+(s1-b)-(a-b) by REAL_1:49; then A35: a-b<=(s1-b)+(s1-b)-(a-b) by XCMPLX_1:26; b-b>=s1-b by A1,REAL_1:49; then b-b+(b-b)>=(s1-b)+(s1-b) by REAL_1:55; then b-b+(b-b)-(a-b)>=(s1-b)+(s1-b)-(a-b) by REAL_1:49; then 0+(b-b)-(a-b)>=(s1-b)+(s1-b)-(a-b) by XCMPLX_1:14; then 0+0-(a-b)>=(s1-b)+(s1-b)-(a-b) by XCMPLX_1:14; then -(a-b)>=(s1-b)+(s1-b)-(a-b) by XCMPLX_1:150; then b-a >=(s1-b)+(s1-b)-(a-b) by XCMPLX_1:143; then (b-a)/(b-a)>=((s1-b)+(s1-b)-(a-b))/(b-a) by A2,REAL_1:73; hence thesis by A2,A30,A31,A32,A33,A34,A35,REAL_1:73,XCMPLX_1:60; end; A36:now assume rng f2 meets K0; then consider x being set such that A37: x in rng f2 & x in K0 by XBOOLE_0:3; reconsider q=x as Point of TOP-REAL 2 by A37; consider p such that A38: p=q &( -1<p`1 & p`1<1 & -1<p`2 & p`2<1) by A37; consider z being set such that A39: z in dom f2 & x=f2.z by A37,FUNCT_1:def 5; reconsider u=z as Point of I[01] by A39; reconsider t=f.u as Point of TOP-REAL 2; A40:ff.t=|[A*(t`1)+B,C*(t`2)+D]| by Def2; ff.t=p by A38,A39,FUNCT_1:22; then A41: -1<A*(t`1)+B & A*(t`1)+B<1 & -1<C*(t`2)+D & C*(t`2)+D<1 by A38,A40,EUCLID:56; A42: A*(t`1)+B=(2*((t`1)-b)-(a-b))/(b-a) proof thus A*(t`1)+B= (t`1)*2/(b-a)+(1-2*b/(b-a)) by XCMPLX_1:75 .= (t`1)*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A2,XCMPLX_1:60 .= (t`1)*2/(b-a)+((b-a)-2*b)/(b-a) by XCMPLX_1:121 .= ((t`1)*2+((b-a)-2*b))/(b-a) by XCMPLX_1:63 .= (2*(t`1)+(b-a)-2*b)/(b-a) by XCMPLX_1:29 .= (2*(t`1)-2*b+(b-a))/(b-a) by XCMPLX_1:29 .= (2*((t`1)-b)+(b-a))/(b-a) by XCMPLX_1:40 .= (2*((t`1)-b)-(a-b))/(b-a) by XCMPLX_1:38; end; then (-1)*(b-a)< (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A2,A41,REAL_1:70; then (-1)*(b-a)< 2*((t`1)-b)-(a-b) by A2,XCMPLX_1:88; then (-1)*(b-a)+(a-b)< 2*((t`1)-b)-(a-b)+(a-b) by REAL_1:67; then (-1)*(b-a)+(a-b)< 2*((t`1)-b) by XCMPLX_1:27; then -(b-a)+(a-b)< 2*((t`1)-b) by XCMPLX_1:180; then a-b+(a-b)< 2*((t`1)-b) by XCMPLX_1:143; then 2*(a-b)< 2*((t`1)-b) by XCMPLX_1:11; then 2*(a-b)/2< 2*((t`1)-b)/2 by REAL_1:73; then (a-b)< ((t`1)-b)*2/2 by XCMPLX_1:90; then a-b < (t`1)-b by XCMPLX_1:90; then A43:a < (t`1) by REAL_1:49; (1)*(b-a)> (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A2,A41,A42,REAL_1:70; then (1)*(b-a)> 2*((t`1)-b)-(a-b) by A2,XCMPLX_1:88; then (1)*(b-a)+(a-b)> 2*((t`1)-b)-(a-b)+(a-b) by REAL_1:67; then (1)*(b-a)+(a-b)> 2*((t`1)-b) by XCMPLX_1:27; then b-a-(b-a)> 2*((t`1)-b) by XCMPLX_1:38; then 0>2*((t`1)-b) by XCMPLX_1:14; then 0/2>((t`1)-b)*2/2 by REAL_1:73; then 0/2>((t`1)-b) by XCMPLX_1:90; then A44: 0+b>t`1 by REAL_1:84; A45: C*(t`2)+D= (t`2)*2/(d-c)+(1-2*d/(d-c)) by XCMPLX_1:75 .= (t`2)*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A3,XCMPLX_1:60 .= (t`2)*2/(d-c)+((d-c)-2*d)/(d-c) by XCMPLX_1:121 .= ((t`2)*2+((d-c)-2*d))/(d-c) by XCMPLX_1:63 .= (2*(t`2)+(d-c)-2*d)/(d-c) by XCMPLX_1:29 .= (2*(t`2)-2*d+(d-c))/(d-c) by XCMPLX_1:29 .= (2*((t`2)-d)+(d-c))/(d-c) by XCMPLX_1:40 .= (2*((t`2)-d)-(c-d))/(d-c) by XCMPLX_1:38; then (-1)*(d-c)< (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A3,A41,REAL_1:70; then (-1)*(d-c)< 2*((t`2)-d)-(c-d) by A3,XCMPLX_1:88; then (-1)*(d-c)+(c-d)< 2*((t`2)-d)-(c-d)+(c-d) by REAL_1:67; then (-1)*(d-c)+(c-d)< 2*((t`2)-d) by XCMPLX_1:27; then -(d-c)+(c-d)< 2*((t`2)-d) by XCMPLX_1:180; then c-d+(c-d)< 2*((t`2)-d) by XCMPLX_1:143; then 2*(c-d)< 2*((t`2)-d) by XCMPLX_1:11; then 2*(c-d)/2< 2*((t`2)-d)/2 by REAL_1:73; then (c-d)< ((t`2)-d)*2/2 by XCMPLX_1:90; then c-d < (t`2)-d by XCMPLX_1:90; then A46:c < (t`2) by REAL_1:49; (1)*(d-c)> (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A3,A41,A45,REAL_1:70; then (1)*(d-c)> 2*((t`2)-d)-(c-d) by A3,XCMPLX_1:88; then (1)*(d-c)+(c-d)> 2*((t`2)-d)-(c-d)+(c-d) by REAL_1:67; then (1)*(d-c)+(c-d)> 2*((t`2)-d) by XCMPLX_1:27; then d-c-(d-c)> 2*((t`2)-d) by XCMPLX_1:38; then 0>2*((t`2)-d) by XCMPLX_1:14; then 0/2>((t`2)-d)*2/2 by REAL_1:73; then 0/2>((t`2)-d) by XCMPLX_1:90; then 0+d>t`2 by REAL_1:84; hence contradiction by A1,A43,A44,A46; end; now assume rng g2 meets K0; then consider x being set such that A47: x in rng g2 & x in K0 by XBOOLE_0:3; reconsider q=x as Point of TOP-REAL 2 by A47; consider p such that A48: p=q &( -1<p`1 & p`1<1 & -1<p`2 & p`2<1) by A47; consider z being set such that A49: z in dom g2 & x=g2.z by A47,FUNCT_1:def 5; reconsider u=z as Point of I[01] by A49; reconsider t=g.u as Point of TOP-REAL 2; A50:ff.t=|[A*(t`1)+B,C*(t`2)+D]| by Def2; ff.t=p by A48,A49,FUNCT_1:22; then A51: -1<A*(t`1)+B & A*(t`1)+B<1 & -1<C*(t`2)+D & C*(t`2)+D<1 by A48,A50,EUCLID:56; A52: A*(t`1)+B= (t`1)*2/(b-a)+(1-2*b/(b-a)) by XCMPLX_1:75 .= (t`1)*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A2,XCMPLX_1:60 .= (t`1)*2/(b-a)+((b-a)-2*b)/(b-a) by XCMPLX_1:121 .= ((t`1)*2+((b-a)-2*b))/(b-a) by XCMPLX_1:63 .= (2*(t`1)+(b-a)-2*b)/(b-a) by XCMPLX_1:29 .= (2*(t`1)-2*b+(b-a))/(b-a) by XCMPLX_1:29 .= (2*((t`1)-b)+(b-a))/(b-a) by XCMPLX_1:40 .= (2*((t`1)-b)-(a-b))/(b-a) by XCMPLX_1:38; then (-1)*(b-a)< (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A2,A51,REAL_1:70; then (-1)*(b-a)< 2*((t`1)-b)-(a-b) by A2,XCMPLX_1:88; then (-1)*(b-a)+(a-b)< 2*((t`1)-b)-(a-b)+(a-b) by REAL_1:67; then (-1)*(b-a)+(a-b)< 2*((t`1)-b) by XCMPLX_1:27; then -(b-a)+(a-b)< 2*((t`1)-b) by XCMPLX_1:180; then a-b+(a-b)< 2*((t`1)-b) by XCMPLX_1:143; then 2*(a-b)< 2*((t`1)-b) by XCMPLX_1:11; then 2*(a-b)/2< 2*((t`1)-b)/2 by REAL_1:73; then (a-b)< ((t`1)-b)*2/2 by XCMPLX_1:90; then a-b < (t`1)-b by XCMPLX_1:90; then A53:a < (t`1) by REAL_1:49; (1)*(b-a)> (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A2,A51,A52,REAL_1:70; then (1)*(b-a)> 2*((t`1)-b)-(a-b) by A2,XCMPLX_1:88; then (1)*(b-a)+(a-b)> 2*((t`1)-b)-(a-b)+(a-b) by REAL_1:67; then (1)*(b-a)+(a-b)> 2*((t`1)-b) by XCMPLX_1:27; then b-a-(b-a)> 2*((t`1)-b) by XCMPLX_1:38; then 0>2*((t`1)-b) by XCMPLX_1:14; then 0/2>((t`1)-b)*2/2 by REAL_1:73; then 0/2>((t`1)-b) by XCMPLX_1:90; then A54: 0+b>t`1 by REAL_1:84; A55: C*(t`2)+D= (t`2)*2/(d-c)+(1-2*d/(d-c)) by XCMPLX_1:75 .= (t`2)*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A3,XCMPLX_1:60 .= (t`2)*2/(d-c)+((d-c)-2*d)/(d-c) by XCMPLX_1:121 .= ((t`2)*2+((d-c)-2*d))/(d-c) by XCMPLX_1:63 .= (2*(t`2)+(d-c)-2*d)/(d-c) by XCMPLX_1:29 .= (2*(t`2)-2*d+(d-c))/(d-c) by XCMPLX_1:29 .= (2*((t`2)-d)+(d-c))/(d-c) by XCMPLX_1:40 .= (2*((t`2)-d)-(c-d))/(d-c) by XCMPLX_1:38; then (-1)*(d-c)< (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A3,A51,REAL_1:70; then (-1)*(d-c)< 2*((t`2)-d)-(c-d) by A3,XCMPLX_1:88; then (-1)*(d-c)+(c-d)< 2*((t`2)-d)-(c-d)+(c-d) by REAL_1:67; then (-1)*(d-c)+(c-d)< 2*((t`2)-d) by XCMPLX_1:27; then -(d-c)+(c-d)< 2*((t`2)-d) by XCMPLX_1:180; then c-d+(c-d)< 2*((t`2)-d) by XCMPLX_1:143; then 2*(c-d)< 2*((t`2)-d) by XCMPLX_1:11; then 2*(c-d)/2< 2*((t`2)-d)/2 by REAL_1:73; then (c-d)< ((t`2)-d)*2/2 by XCMPLX_1:90; then c-d < (t`2)-d by XCMPLX_1:90; then A56:c < (t`2) by REAL_1:49; (1)*(d-c)> (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A3,A51,A55,REAL_1:70; then (1)*(d-c)> 2*((t`2)-d)-(c-d) by A3,XCMPLX_1:88; then (1)*(d-c)+(c-d)> 2*((t`2)-d)-(c-d)+(c-d) by REAL_1:67; then (1)*(d-c)+(c-d)> 2*((t`2)-d) by XCMPLX_1:27; then d-c-(d-c)> 2*((t`2)-d) by XCMPLX_1:38; then 0>2*((t`2)-d) by XCMPLX_1:14; then 0/2>((t`2)-d)*2/2 by REAL_1:73; then 0/2>((t`2)-d) by XCMPLX_1:90; then 0+d>t`2 by REAL_1:84; hence contradiction by A1,A53,A54,A56; end; then rng f2 meets rng g2 by A1,A10,A11,A12,A13,A18,A19,A20,A21,A22,A29,A36, Th52; then A57:rng f2 /\ rng g2 <> {} by XBOOLE_0:def 7; consider y being Element of rng f2 /\ rng g2; A58: y in rng f2 & y in rng g2 by A57,XBOOLE_0:def 3; then consider x being set such that A59: x in dom f2 & y=f2.x by FUNCT_1:def 5; A60: y=ff.(f.x) by A59,FUNCT_1:22; consider x2 being set such that A61: x2 in dom g2 & y=g2.x2 by A58,FUNCT_1:def 5; A62: y=ff.(g.x2) by A61,FUNCT_1:22; dom f2 c= dom f by RELAT_1:44; then A63: f.x in rng f by A59,FUNCT_1:12; dom g2 c= dom g by RELAT_1:44; then A64: g.x2 in rng g by A61,FUNCT_1:12; then f.x=g.x2 by A6,A9,A60,A62,A63,FUNCT_1:def 8; then rng f /\ rng g <> {} by A63,A64,XBOOLE_0:def 3; hence thesis by XBOOLE_0:def 7; end; theorem {p7 where p7 is Point of TOP-REAL 2: p7`2<=p7`1 } is closed Subset of TOP-REAL 2 & {p7 where p7 is Point of TOP-REAL 2: p7`1<=p7`2 } is closed Subset of TOP-REAL 2 by Lm5,Lm8; theorem {p7 where p7 is Point of TOP-REAL 2: -p7`1<=p7`2 } is closed Subset of TOP-REAL 2 & {p7 where p7 is Point of TOP-REAL 2: p7`2<=-p7`1 } is closed Subset of TOP-REAL 2 by Lm11,Lm14; theorem {p7 where p7 is Point of TOP-REAL 2: -p7`2<=p7`1 } is closed Subset of TOP-REAL 2 & {p7 where p7 is Point of TOP-REAL 2: p7`1<=-p7`2 } is closed Subset of TOP-REAL 2 by Lm17,Lm20;

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