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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