The Mizar article:

The Topological Space $\calE^2_\rmT$. Simple Closed Curves

by
Agata Darmochwal, and
Yatsuka Nakamura

Received December 30, 1991

Copyright (c) 1991 Association of Mizar Users

MML identifier: TOPREAL2
[ MML identifier index ]


environ

 vocabulary PRE_TOPC, EUCLID, FINSEQ_1, TOPREAL1, MCART_1, BOOLE, COMPTS_1,
      BORSUK_1, RELAT_1, TOPS_2, ORDINAL2, SUBSET_1, FUNCT_1, RCOMP_1,
      PCOMPS_1, FUNCT_4, TOPREAL2, FINSEQ_4;
 notation TARSKI, XBOOLE_0, SUBSET_1, XREAL_0, RELAT_1, FUNCT_1, FUNCT_2,
      FUNCT_4, STRUCT_0, GRCAT_1, PRE_TOPC, TOPS_2, COMPTS_1, RCOMP_1,
      FINSEQ_1, FINSEQ_4, PCOMPS_1, EUCLID, TOPMETR, TOPREAL1;
 constructors TOPS_2, COMPTS_1, RCOMP_1, TOPMETR, TOPREAL1, FINSEQ_4, GRCAT_1,
      XCMPLX_0, MEMBERED, XBOOLE_0;
 clusters SUBSET_1, FUNCT_1, PRE_TOPC, RELSET_1, STRUCT_0, EUCLID, BORSUK_1,
      TOPREAL1, XREAL_0, XBOOLE_0, MEMBERED, ZFMISC_1;
 requirements NUMERALS, REAL, BOOLE, SUBSET;
 definitions TARSKI, PRE_TOPC, TOPS_2;
 theorems AXIOMS, BORSUK_1, COMPTS_1, ENUMSET1, EUCLID, FUNCT_1, FUNCT_2,
      HEINE, PCOMPS_1, PRE_TOPC, RCOMP_1, REAL_1, TARSKI, TOPMETR, TOPMETR2,
      TOPS_2, ZFMISC_1, TOPREAL1, RELAT_1, RELSET_1, GRCAT_1, XBOOLE_0,
      XBOOLE_1;

begin

reserve a for set;
reserve p,p1,p2,q,q1,q2 for Point of TOP-REAL 2;
reserve h1,h2 for FinSequence of TOP-REAL 2;

Lm1:
  now let x,X be set; assume not x in X;
   then {x} misses X by ZFMISC_1:56;
   hence {x} /\ X = {} by XBOOLE_0:def 7;
  end;

Lm2:
 LSeg(|[0,0]|,|[1,0]|) /\ LSeg(|[0,1]|,|[1,1]|) = {}
   by TOPREAL1:25,XBOOLE_0:def 7;

Lm3:
 LSeg(|[0,0]|,|[0,1]|) /\ LSeg(|[1,0]|,|[1,1]|) = {}
   by TOPREAL1:26,XBOOLE_0:def 7;

 set p00 = |[ 0,0 ]|, p01 = |[ 0,1 ]|, p10 = |[ 1,0 ]|, p11 = |[ 1,1 ]|,
  L1 = LSeg(p00,p01),L2 = LSeg(p01,p11),L3 = LSeg(p00,p10),L4 = LSeg(p10,p11);

Lm4: p00`1 = 0 & p00`2 = 0 & p01`1 = 0 & p01`2 = 1 & p10`1 = 1 & p10`2 = 0 &
    p11`1 = 1 & p11`2 = 1 by EUCLID:56;

then Lm5: not p00 in L4 by TOPREAL1:9;

Lm6: not p00 in L2 by Lm4,TOPREAL1:10;

Lm7: not p01 in L3 by Lm4,TOPREAL1:10;

Lm8: not p01 in L4 by Lm4,TOPREAL1:9;

Lm9: not p10 in L1 by Lm4,TOPREAL1:9;

Lm10: not p10 in L2 by Lm4,TOPREAL1:10;

Lm11: not p11 in L1 by Lm4,TOPREAL1:9;

Lm12: not p11 in L3 by Lm4,TOPREAL1:10;

Lm13: p00 in L1 by TOPREAL1:6;

Lm14: p00 in L3 by TOPREAL1:6;

Lm15: p01 in L1 by TOPREAL1:6;

Lm16: p01 in L2 by TOPREAL1:6;

Lm17: p10 in L3 by TOPREAL1:6;

Lm18: p10 in L4 by TOPREAL1:6;

Lm19: p11 in L2 by TOPREAL1:6;

Lm20: p11 in L4 by TOPREAL1:6;

set L = { p : p`1 = 0 & p`2 <= 1 & p`2 >= 0 or
               p`1 <= 1 & p`1 >= 0 & p`2 = 1 or
               p`1 <= 1 & p`1 >= 0 & p`2 = 0 or
               p`1 = 1 & p`2 <= 1 & p`2 >= 0 };
Lm21:p00 in L by Lm4;

Lm22:p11 in L by Lm4;

Lm23:
 p1 <> p2 & p2 in R^2-unit_square &
 p1 in LSeg(p00, p01) implies
  ex P1,P2 being non empty Subset of TOP-REAL 2
  st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
   R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2}
 proof assume that
 A1: p1 <> p2 and
 A2: p2 in R^2-unit_square and
 A3: p1 in LSeg(p00, p01);
A4: p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:20,XBOOLE_0:def 2;
   consider p such that
  A5: p = p1 and A6: p`1 = 0 & p`2 <= 1 & p`2 >= 0 by A3,TOPREAL1:19;
  A7: LSeg(p00,p1) c= L1 by A3,Lm13,TOPREAL1:12;
  A8: LSeg(p1,p01) c= L1 by A3,Lm15,TOPREAL1:12;
   A9: L1 /\ L4 = {} by TOPREAL1:26,XBOOLE_0:def 7;
        LSeg(p1,p00) /\ L4 c= L1 /\ L4 by A7,XBOOLE_1:26;
  then A10: LSeg(p1,p00) /\ L4 = {} by A9,XBOOLE_1:3;
        p00 in LSeg(p1,p00) by TOPREAL1:6;
      then p00 in LSeg(p1,p00) /\ L3 by Lm14,XBOOLE_0:def 3;
  then A11:  {p00} c= LSeg(p1,p00) /\ L3 by ZFMISC_1:37;
      A12: LSeg(p1,p00) /\ L3 c= L1 /\ L3 & L1 /\ L3 = {p00}
       by A7,TOPREAL1:23,XBOOLE_1:26;
  then A13: LSeg(p1,p00) /\ L3 = {p00} by A11,XBOOLE_0:def 10;
        LSeg(p01,p1) /\ L4 c= L1 /\ L4 by A8,XBOOLE_1:26;
  then A14: LSeg(p01,p1) /\ L4 = {} by A9,XBOOLE_1:3;
        p01 in LSeg(p01,p1) by TOPREAL1:6;
      then p01 in LSeg(p01,p1) /\ L2 by Lm16,XBOOLE_0:def 3;
   then A15: {p01} c= LSeg(p01,p1) /\ L2 by ZFMISC_1:37;
A16:      LSeg(p01,p1) /\ L2 c= {p01} by A8,TOPREAL1:21,XBOOLE_1:26;
    per cases by A4,XBOOLE_0:def 2;
    suppose A17: p2 in L1;
     then consider q such that
  A18: q = p2 and A19: q`1 = 0 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:19;
 A20: LSeg(p2,p1) c= L1 by A3,A17,TOPREAL1:12;
A21:      p = |[p`1,p`2]| & q = |[q`1,q`2]| by EUCLID:57;
       now per cases by A1,A5,A6,A18,A19,A21,REAL_1:def 5;
      case A22: p`2 < q`2;
    A23: LSeg(p01,p2) c= L1 by A17,Lm15,TOPREAL1:12;
        then A24: LSeg(p01,p2) /\ L3 c= {p00} by TOPREAL1:23,XBOOLE_1:26;
      set P1 = LSeg(p1,p2),
          P2 = LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2));
          now assume p00 in LSeg(p01,p2) /\ L3;
         then p00 in
 LSeg(p2,p01) & p2`2 <= p01`2 by A18,A19,EUCLID:56,XBOOLE_0:def 3;
         hence contradiction by A6,A18,A22,Lm4,TOPREAL1:10;
        end;
        then {p00} <> LSeg(p01,p2) /\ L3 by ZFMISC_1:37;
    then A25: LSeg(p01,p2) /\ L3 = {} by A24,ZFMISC_1:39;
    A26: LSeg(p01,p2) /\ L4 c= L1 /\ L4 by A23,XBOOLE_1:26;
A27:    now assume
A28:      LSeg(p1,p00) /\ LSeg(p01,p2) <> {};
         consider a being Element of LSeg(p1,p00) /\ LSeg(p01,p2);
         reconsider p = a as Point of TOP-REAL 2 by A28,TARSKI:def 3;
           p in LSeg(p00,p1) & p in LSeg(p2,p01) & p00`2 <= p1`2 &
          p2`2 <= p01`2 by A5,A6,A18,A19,A28,EUCLID:56,XBOOLE_0:def 3;
         then p`2 <= p1`2 & p2`2 <= p`2 by TOPREAL1:10;
         hence contradiction by A5,A18,A22,AXIOMS:22;
        end;
          p01 in LSeg(p01,p2) by TOPREAL1:6;
        then p01 in LSeg(p01,p2) /\ L2 by Lm16,XBOOLE_0:def 3;
    then A29:  {p01} c= LSeg(p01,p2) /\ L2 by ZFMISC_1:37;
A30:        LSeg(p01,p2) /\ L2 c= L1 /\ L2 & L1 /\ L2 = {p01}
         by A23,TOPREAL1:21,XBOOLE_1:26;
    A31:  LSeg(p1,p00) /\ L2 c= {p01} by A7,TOPREAL1:21,XBOOLE_1:26;
          now assume p01 in LSeg(p1,p00) /\ L2;
         then p01 in LSeg(p00,p1) & p00`2 <=
 p1`2 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         then p01`2 <= p1`2 by TOPREAL1:10;
         hence contradiction by A5,A6,A19,A22,Lm4,AXIOMS:21;
        end;
then A32:        {p01} <> LSeg(p1,p00) /\ L2 by ZFMISC_1:37;
        A33: LSeg(p01,p2) \/ LSeg(p2,p1) \/
 LSeg(p1,p00) = L1 by A3,A17,TOPREAL1:13;
   A34: LSeg(p1,p2) c= L1 by A3,A17,TOPREAL1:12;
   then A35: LSeg(p1,p2) /\ L4 c= L1 /\ L4 by XBOOLE_1:26;
          p1 in LSeg(p1,p2) & p1 in LSeg(p1,p00) by TOPREAL1:6;
        then p1 in LSeg(p1,p2) /\ LSeg(p1,p00) by XBOOLE_0:def 3;
    then A36:  {p1} c= LSeg(p1,p2) /\ LSeg(p1,p00) by ZFMISC_1:37;
          LSeg(p1,p2) /\ LSeg(p1,p00) c= {p1}
         proof
          let a; assume A37: a in LSeg(p1,p2) /\ LSeg(p1,p00);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p1,p2) & p in LSeg(p00,p1) & p1`2 <= p2`2 & p1`1 <= p2`1
           & p00`2 <= p1`2 by A5,A6,A18,A19,A22,A37,EUCLID:56,XBOOLE_0:def 3;
          then p1`1 <= p`1 & p`1 <= p2`1 & p1`2 <= p`2 & p`2 <= p1`2
           by TOPREAL1:9,10;
          then p1`2 = p`2 & p`1 = 0 by A5,A6,A18,A19,AXIOMS:21;
          then p = |[ 0, p1`2]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
          hence a in {p1} by TARSKI:def 1;
         end;
   then A38: LSeg(p1,p2) /\ LSeg(p1,p00) = {p1} by A36,XBOOLE_0:def 10;
          p2 in LSeg(p1,p2) & p2 in LSeg(p01,p2) by TOPREAL1:6;
        then p2 in LSeg(p1,p2) /\ LSeg(p01,p2) by XBOOLE_0:def 3;
    then A39:  {p2} c= LSeg(p1,p2) /\ LSeg(p01,p2) by ZFMISC_1:37;
          LSeg(p1,p2) /\ LSeg(p01,p2) c= {p2}
         proof
          let a; assume A40: a in LSeg(p1,p2) /\ LSeg(p01,p2);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p1,p2) & p in LSeg(p2,p01) & p1`2 <= p2`2 & p1`1 <= p2`1
           & p2`2 <= p01`2 by A5,A6,A18,A19,A22,A40,EUCLID:56,XBOOLE_0:def 3;
          then p1`1 <= p`1 & p`1 <= p2`1 & p2`2 <= p`2 & p`2 <= p2`2
           by TOPREAL1:9,10;
          then p2`2 = p`2 & p`1 = 0 by A5,A6,A18,A19,AXIOMS:21;
          then p = |[0,p2`2]| by EUCLID:57 .= p2 by A18,A19,EUCLID:57;
          hence a in {p2} by TARSKI:def 1;
         end;
then A41:  LSeg(p1,p2) /\ LSeg(p01,p2) = {p2} by A39,XBOOLE_0:def 10;
     thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
     A42: L3 \/ L4 is_an_arc_of p00,p11 by Lm4,TOPREAL1:18,22;
       (L3 \/ L4) /\ LSeg(p11,p01)
      = {} \/ {p11} by Lm2,TOPREAL1:24,XBOOLE_1:23 .= {p11};
     then A43: L3 \/ L4 \/ LSeg(p11,p01) is_an_arc_of p00,p01 by A42,TOPREAL1:
16;
         (L3 \/ L4 \/ LSeg(p11,p01)) /\ LSeg(p01,p2)
       = LSeg(p01,p2) /\ (L3 \/ L4) \/ (LSeg(p01,p2) /\
 LSeg(p11,p01)) by XBOOLE_1:23
      .= {}
 \/ (LSeg(p01,p2) /\ L4) \/ (LSeg(p01,p2) /\ LSeg(p11,p01)) by A25,XBOOLE_1:23
      .= {} \/ (LSeg(p01,p2) /\ LSeg(p11,p01)) by A26,Lm3,XBOOLE_1:3
      .= {p01} by A29,A30,XBOOLE_0:def 10;
     then A44: L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2) is_an_arc_of p00,p2
       by A43,TOPREAL1:16;
          LSeg(p1,p00) /\ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2))
         = LSeg(p1,p00) /\ (L3 \/ L4 \/ LSeg(p11,p01))
            \/ LSeg(p1,p00) /\ LSeg(p01,p2) by XBOOLE_1:23
        .= LSeg(p1,p00) /\ (L3 \/ L4)
            \/ LSeg(p1,p00) /\ LSeg(p11,p01)
            \/ LSeg(p1,p00) /\ LSeg(p01,p2) by XBOOLE_1:23
        .= (LSeg(p1,p00) /\ L3) \/ (LSeg(p1,p00) /\ L4)
            \/ (LSeg(p1,p00) /\ LSeg(p11,p01))
            \/ (LSeg(p1,p00) /\ LSeg(p01,p2)) by XBOOLE_1:23
        .= {p00} by A10,A13,A27,A31,A32,ZFMISC_1:39;
      hence LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2))
       is_an_arc_of p1,p2 by A44,TOPREAL1:17;
      thus R^2-unit_square =
LSeg(p1,p2) \/ LSeg(p01,p2) \/ LSeg(p1,p00) \/ (L3 \/ L4 \/
 LSeg(p11,p01)) by A33,TOPREAL1:20,XBOOLE_1:4
       .= LSeg(p1,p2) \/ (LSeg(p1,p00) \/ LSeg(p01,p2))
           \/ (L3 \/ L4 \/ LSeg(p11,p01)) by XBOOLE_1:4
       .= LSeg(p1,p2) \/ ((LSeg(p1,p00) \/ LSeg(p01,p2))
           \/ (L3 \/ L4 \/ LSeg(p11,p01))) by XBOOLE_1:4
       .= P1 \/ P2 by XBOOLE_1:4;
 A45:   P1 /\ P2
      = (LSeg(p1,p2) /\ LSeg(p1,p00))
         \/ LSeg(p1,p2) /\ (L3 \/ L4 \/ LSeg(p11,p01) \/
 LSeg(p01,p2)) by XBOOLE_1:23
     .= (LSeg(p1,p2) /\ LSeg(p1,p00))
         \/ ((LSeg(p1,p2) /\ (L3 \/ L4 \/ LSeg(p11,p01)))
         \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by XBOOLE_1:23
     .= (LSeg(p1,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p1,p2) /\ (L3 \/ L4))
         \/ (LSeg(p1,p2) /\ LSeg(p11,p01))
         \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ (LSeg(p1,p2) /\ L4)
         \/ (LSeg(p1,p2) /\ LSeg(p11,p01))
         \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by A38,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ {} \/ (LSeg(p1,p2) /\ LSeg(p11,p01))
         \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by A35,Lm3,XBOOLE_1:3
     .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/ {p2})) by A41,
XBOOLE_1:4
     .= {p1} \/ (LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/
 {p2}) by XBOOLE_1:4;
A46:   LSeg(p1,p2) /\ L3 c= {p00} by A34,TOPREAL1:23,XBOOLE_1:26;
A47:    now per cases;
      suppose A48: p1 = p00;
       then p00 in LSeg(p1,p2) by TOPREAL1:6;
       then LSeg(p1,p2) /\ L3 <> {} by Lm14,XBOOLE_0:def 3;
       then LSeg(p1,p2) /\ L3 = {p1} by A46,A48,ZFMISC_1:39;
       hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A45;
      suppose A49: p1 <> p00;
         now assume p00 in LSeg(p1,p2) /\ L3;
        then p00 in LSeg(p1,p2) & p1`2 <= p2`2 & p00`2 <= p01`2
         by A5,A18,A22,Lm4,XBOOLE_0:def 3;
        then p00`2 <= p1`2 & p1`2 <= p00`2 by A3,TOPREAL1:10;
        then p00`2 = p1`2 by AXIOMS:21;
        hence contradiction by A5,A6,A49,Lm4,EUCLID:57;
       end;
       then LSeg(p1,p2) /\ L3 <> {p00} by ZFMISC_1:37;
       then LSeg(p1,p2) /\ L3 = {} by A46,ZFMISC_1:39;
       hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A45;
      end;
A50:    LSeg(p1,p2) /\ L2 c= {p01} by A34,TOPREAL1:21,XBOOLE_1:26;
        now per cases;
       suppose A51: p2 <> p01;
         now assume p01 in LSeg(p1,p2) /\ L2;
        then p01 in LSeg(p1,p2) & p1`2 <= p2`2 & p00`2 <= p01`2
         by A5,A18,A22,Lm4,XBOOLE_0:def 3;
        then p01`2 <= p2`2 & p2`2 <= p01`2 by A17,TOPREAL1:10;
        then A52: p01`2 = p2`2 by AXIOMS:21;
          p2 = |[p2`1,p2`2]| by EUCLID:57 .= |[0,1]| by A18,A19,A52,EUCLID:56;
        hence contradiction by A51;
       end;
       then LSeg(p1,p2) /\ L2 <> {p01} by ZFMISC_1:37;
       then LSeg(p1,p2) /\ L2 = {} by A50,ZFMISC_1:39;
       hence P1 /\ P2 = {p1,p2} by A47,ENUMSET1:41;
       suppose A53: p2 = p01;
        then p01 in LSeg(p1,p2) by TOPREAL1:6;
        then LSeg(p1,p2) /\ L2 <> {} by Lm16,XBOOLE_0:def 3;
       then LSeg(p1,p2) /\ L2 = {p2} by A50,A53,ZFMISC_1:39;
       hence P1 /\ P2 = {p1,p2} by A47,ENUMSET1:41;
      end;
      hence P1 /\ P2 = {p1,p2};
      case A54: p`2 > q`2;
       take P1 = LSeg(p2,p1),P2
 = LSeg(p2,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p1));
        A55: LSeg(p01,p1) /\ L3 c= {p00} by A8,TOPREAL1:23,XBOOLE_1:26;
          now assume p00 in LSeg(p01,p1) /\ L3;
         then p00 in LSeg(p1,p01) & p1`2 <=
 p01`2 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         hence contradiction by A5,A19,A54,Lm4,TOPREAL1:10;
        end;
then A56:        {p00} <> LSeg(p01,p1) /\ L3 by ZFMISC_1:37;
    A57: LSeg(p2,p00) c= L1 by A17,Lm13,TOPREAL1:12;
    then A58: LSeg(p2,p00) /\ L4 c= L1 /\ L4 by XBOOLE_1:26;
A59:        now assume
A60:         LSeg(p2,p00) /\ LSeg(p01,p1) <> {};
         consider a being Element of LSeg(p2,p00) /\ LSeg(p01,p1);
         reconsider p = a as Point of TOP-REAL 2 by A60,TARSKI:def 3;
           p in LSeg(p00,p2) & p in LSeg(p1,p01) & p00`2 <= p2`2 &
          p1`2 <= p01`2 by A5,A6,A18,A19,A60,EUCLID:56,XBOOLE_0:def 3;
         then p`2 <= p2`2 & p1`2 <= p`2 by TOPREAL1:10;
         hence contradiction by A5,A18,A54,AXIOMS:22;
        end;
          p00 in LSeg(p2,p00) by TOPREAL1:6;
        then p00 in LSeg(p2,p00) /\ L3 by Lm14,XBOOLE_0:def 3;
    then A61:  {p00} c= LSeg(p2,p00) /\ L3 by ZFMISC_1:37;
A62:        LSeg(p2,p00) /\ L3 c= {p00} by A57,TOPREAL1:23,XBOOLE_1:26;
    A63:  LSeg(p2,p00) /\ L2 c= {p01} by A57,TOPREAL1:21,XBOOLE_1:26;
          now assume p01 in LSeg(p2,p00) /\ L2;
         then p01 in
 LSeg(p00,p2) & p00`2 <= p2`2 by A18,A19,EUCLID:56,XBOOLE_0:def 3;
         then p01`2 <= p2`2 by TOPREAL1:10;
         hence contradiction by A6,A18,A19,A54,Lm4,AXIOMS:21;
        end;
then A64:        {p01} <> LSeg(p2,p00) /\ L2 by ZFMISC_1:37;
        A65: LSeg(p01,p1) \/ LSeg(p1,p2) \/
 LSeg(p2,p00) = L1 by A3,A17,TOPREAL1:13;
   A66: LSeg(p2,p1) /\ L4 c= L1 /\ L4 by A20,XBOOLE_1:26;
          p2 in LSeg(p2,p1) & p2 in LSeg(p2,p00) by TOPREAL1:6;
        then p2 in LSeg(p2,p1) /\ LSeg(p2,p00) by XBOOLE_0:def 3;
    then A67:  {p2} c= LSeg(p2,p1) /\ LSeg(p2,p00) by ZFMISC_1:37;
A68:        LSeg(p2,p1) /\ LSeg(p2,p00) c= {p2}
         proof
          let a; assume A69: a in LSeg(p2,p1) /\ LSeg(p2,p00);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p2,p1) & p in LSeg(p00,p2) & p2`2 <= p1`2 & p2`1 <= p1`1
           & p00`2 <= p2`2 by A5,A6,A18,A19,A54,A69,EUCLID:56,XBOOLE_0:def 3;
          then p2`1 <= p`1 & p`1 <= p1`1 & p2`2 <= p`2 & p`2 <= p2`2
           by TOPREAL1:9,10;
          then p2`2 = p`2 & p`1 = 0 by A5,A6,A18,A19,AXIOMS:21;
          then p = |[ 0, p2`2]| by EUCLID:57 .= p2 by A18,A19,EUCLID:57;
          hence a in {p2} by TARSKI:def 1;
         end;
          p1 in LSeg(p2,p1) & p1 in LSeg(p01,p1) by TOPREAL1:6;
        then p1 in LSeg(p2,p1) /\ LSeg(p01,p1) by XBOOLE_0:def 3;
    then A70:  {p1} c= LSeg(p2,p1) /\ LSeg(p01,p1) by ZFMISC_1:37;
A71:        LSeg(p2,p1) /\ LSeg(p01,p1) c= {p1}
         proof
          let a; assume A72: a in LSeg(p2,p1) /\ LSeg(p01,p1);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p2,p1) & p in LSeg(p1,p01) & p2`2 <= p1`2 & p2`1 <= p1`1
           & p1`2 <= p01`2 by A5,A6,A18,A19,A54,A72,EUCLID:56,XBOOLE_0:def 3;
          then p2`1 <= p`1 & p`1 <= p1`1 & p1`2 <= p`2 & p`2 <= p1`2
           by TOPREAL1:9,10;
          then p1`2 = p`2 & p`1 = 0 by A5,A6,A18,A19,AXIOMS:21;
          then p = |[0,p1`2]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
          hence a in {p1} by TARSKI:def 1;
         end;
   thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
A73: L3 \/ L4 is_an_arc_of p11,p00 by Lm4,TOPREAL1:18,22;
       L2 /\ (L3 \/ L4)
      = {} \/ {p11} by Lm2,TOPREAL1:24,XBOOLE_1:23 .= {p11};
     then A74: L3 \/ L4 \/ LSeg(p11,p01) is_an_arc_of p01,p00 by A73,TOPREAL1:
17;
        LSeg(p1,p01) /\ (L3 \/ L4 \/ LSeg(p11,p01))
       = LSeg(p01,p1) /\ (L3 \/ L4) \/ (LSeg(p01,p1) /\
 LSeg(p11,p01)) by XBOOLE_1:23
      .= (LSeg(p01,p1) /\ L3) \/ (LSeg(p01,p1) /\ L4)
           \/ (LSeg(p01,p1) /\ LSeg(p11,p01)) by XBOOLE_1:23
      .= {} \/ (LSeg(p01,p1) /\ L4) \/ (LSeg(p01,p1) /\ LSeg(p11,p01))
        by A55,A56,ZFMISC_1:39
      .= {p01} by A14,A15,A16,XBOOLE_0:def 10;
      then A75: L3 \/ L4 \/ LSeg(p11,p01) \/
 LSeg(p01,p1) is_an_arc_of p1,p00 by A74,TOPREAL1:17;
          (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p1)) /\ LSeg(p00,p2)
         = LSeg(p2,p00) /\ (L3 \/ L4 \/ LSeg(p11,p01))
            \/ LSeg(p2,p00) /\ LSeg(p01,p1) by XBOOLE_1:23
        .= LSeg(p2,p00) /\ (L3 \/ L4)
            \/ LSeg(p2,p00) /\ LSeg(p11,p01)
            \/ LSeg(p2,p00) /\ LSeg(p01,p1) by XBOOLE_1:23
        .= (LSeg(p2,p00) /\ L3) \/ (LSeg(p2,p00) /\ L4)
            \/ (LSeg(p2,p00) /\ LSeg(p11,p01))
            \/ (LSeg(p2,p00) /\ LSeg(p01,p1)) by XBOOLE_1:23
        .= (LSeg(p2,p00) /\ L3) \/ {} \/ (LSeg(p2,p00) /\ LSeg(p11,p01))
            \/ (LSeg(p2,p00) /\ LSeg(p01,p1)) by A58,Lm3,XBOOLE_1:3
        .= LSeg(p2,p00) /\ L3 \/ {} by A59,A63,A64,ZFMISC_1:39
        .= {p00} by A61,A62,XBOOLE_0:def 10;
      hence P2 is_an_arc_of p1,p2 by A75,TOPREAL1:16;
      thus R^2-unit_square =
LSeg(p2,p1) \/ LSeg(p01,p1) \/ LSeg(p2,p00) \/ (L3 \/ L4 \/
 LSeg(p11,p01)) by A65,TOPREAL1:20,XBOOLE_1:4
       .= LSeg(p2,p1) \/ (LSeg(p2,p00) \/ LSeg(p01,p1))
           \/ (L3 \/ L4 \/ LSeg(p11,p01)) by XBOOLE_1:4
       .= LSeg(p2,p1) \/ ((LSeg(p2,p00) \/ LSeg(p01,p1))
           \/ (L3 \/ L4 \/ LSeg(p11,p01))) by XBOOLE_1:4
       .= P1 \/ P2 by XBOOLE_1:4;
 A76:   P1 /\ P2 = (LSeg(p2,p1) /\ LSeg(p2,p00))
         \/ LSeg(p2,p1) /\ (L3 \/ L4 \/ LSeg(p11,p01) \/
 LSeg(p01,p1)) by XBOOLE_1:23
     .= (LSeg(p2,p1) /\ LSeg(p2,p00))
         \/ ((LSeg(p2,p1) /\ (L3 \/ L4 \/ LSeg(p11,p01)))
         \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by XBOOLE_1:23
     .= (LSeg(p2,p1) /\ LSeg(p2,p00)) \/ ((LSeg(p2,p1) /\ (L3 \/ L4))
         \/ (LSeg(p2,p1) /\ LSeg(p11,p01))
         \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by XBOOLE_1:23
     .= (LSeg(p2,p1) /\ LSeg(p2,p00))
         \/ ((LSeg(p2,p1) /\ L3) \/ (LSeg(p2,p1) /\ L4)
         \/ (LSeg(p2,p1) /\ LSeg(p11,p01))
         \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by XBOOLE_1:23
     .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ (LSeg(p2,p1) /\ L4)
         \/ (LSeg(p2,p1) /\ LSeg(p11,p01))
         \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by A67,A68,XBOOLE_0:def 10
     .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ {} \/ (LSeg(p2,p1) /\ LSeg(p11,p01))
         \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by A66,Lm3,XBOOLE_1:3
     .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ LSeg(p2,p1) /\ L2 \/
 {p1}) by A70,A71,XBOOLE_0:def 10
     .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ ((LSeg(p2,p1) /\ L2) \/
 {p1})) by XBOOLE_1:4
     .= {p2} \/ (LSeg(p2,p1) /\ L3) \/ ((LSeg(p2,p1) /\ L2) \/
 {p1}) by XBOOLE_1:4;
A77:   LSeg(p2,p1) /\ L3 c= {p00} by A20,TOPREAL1:23,XBOOLE_1:26;
A78:    now per cases;
       suppose A79: p2 = p00;
         p2 in LSeg(p2,p1) by TOPREAL1:6;
       then LSeg(p2,p1) /\ L3 <> {} by A79,Lm14,XBOOLE_0:def 3;
       then LSeg(p2,p1) /\ L3 = {p2} by A77,A79,ZFMISC_1:39;
       hence P1 /\ P2 = {p2} \/ ((LSeg(p2,p1) /\ L2) \/ {p1}) by A76;
       suppose A80: p2 <> p00;
         now assume p00 in LSeg(p2,p1) /\ L3;
        then p00 in LSeg(p2,p1) & p2`2 <= p1`2 & p00`2 <= p01`2
         by A5,A18,A54,Lm4,XBOOLE_0:def 3;
        then p00`2 <= p2`2 & p2`2 <= p00`2 by A17,TOPREAL1:10;
        then p00`2 = p2`2 by AXIOMS:21;
        hence contradiction by A18,A19,A80,Lm4,EUCLID:57;
       end;
       then LSeg(p2,p1) /\ L3 <> {p00} by ZFMISC_1:37;
       then LSeg(p2,p1) /\ L3 = {} by A77,ZFMISC_1:39;
       hence P1 /\ P2 = {p2} \/ ((LSeg(p2,p1) /\ L2) \/ {p1}) by A76;
      end;
A81:    LSeg(p2,p1) /\ L2 c= {p01} by A20,TOPREAL1:21,XBOOLE_1:26;
        now per cases;
       suppose A82: p1 <> p01;
         now assume p01 in LSeg(p2,p1) /\ L2;
        then p01 in LSeg(p2,p1) & p2`2 <= p1`2 & p00`2 <= p01`2
         by A5,A18,A54,Lm4,XBOOLE_0:def 3;
        then p01`2 <= p1`2 & p1`2 <= p01`2 by A3,TOPREAL1:10;
        then A83: p01`2 = p1`2 by AXIOMS:21;
          p1 = |[p1`1,p1`2]| by EUCLID:57 .= |[0,1]| by A5,A6,A83,EUCLID:56;
        hence contradiction by A82;
       end;
       then LSeg(p2,p1) /\ L2 <> {p01} by ZFMISC_1:37;
       then LSeg(p2,p1) /\ L2 = {} by A81,ZFMISC_1:39;
       hence P1 /\ P2 = {p1,p2} by A78,ENUMSET1:41;
       suppose A84: p1 = p01;
       then p01 in LSeg(p2,p1) by TOPREAL1:6;
       then LSeg(p2,p1) /\ L2 <> {} by Lm16,XBOOLE_0:def 3;
       then LSeg(p2,p1) /\ L2 = {p1} by A81,A84,ZFMISC_1:39;
       hence P1 /\ P2 = {p1,p2} by A78,ENUMSET1:41;
      end;
      hence P1 /\ P2 = {p1,p2};
     end;
    hence ex P1,P2 being non empty Subset of TOP-REAL 2
     st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
     R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2};
    suppose A85: p2 in L2;
     then A86: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 1 by TOPREAL1:19;
    A87: p1 <> p01 or p2 <> p01 by A1;
    take P1 = LSeg(p1,p01) \/ LSeg(p01,p2),P2
 = LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2));
      p01 in LSeg(p1,p01) & p01 in LSeg(p01,p2) by TOPREAL1:6;
    then LSeg(p1,p01) c= L1 & LSeg(p01,p2) c= L2 &
     p01 in LSeg(p1,p01) /\ LSeg(p01,p2) by A3,A85,Lm15,Lm16,TOPREAL1:12,
XBOOLE_0:def 3;
    then LSeg(p1,p01) /\ LSeg(p01,p2) c= L1 /\ L2 & L1 /\ L2 = {p01}
     & LSeg(p1,p01) /\ LSeg(p01,p2) <> {} by TOPREAL1:21,XBOOLE_1:27;
    then LSeg(p1,p01) /\ LSeg(p01,p2) = {p01} by ZFMISC_1:39;
    hence P1 is_an_arc_of p1,p2 by A87,TOPREAL1:18;
      L3 is_an_arc_of p00,p10 & L4 is_an_arc_of p10,p11 by Lm4,TOPREAL1:15;
    then A88: L3 \/ L4 is_an_arc_of p00,p11 by TOPREAL1:5,22;
A89: LSeg(p11,p2) c= L2 by A85,Lm19,TOPREAL1:12;
    then L3 /\ LSeg(p11,p2) c= L3 /\ L2 by XBOOLE_1:26;
then A90: L3 /\ LSeg(p11,p2) = {} by Lm2,XBOOLE_1:3;
      p11 in LSeg(p11,p2) by TOPREAL1:6;
    then A91: L4 /\ LSeg(p11,p2) c= L4 /\ L2 &
     p11 in L4 /\ LSeg(p11,p2) by A89,Lm20,XBOOLE_0:def 3,XBOOLE_1:27;
      (L3 \/ L4) /\ LSeg(p11,p2)
     = (L3 /\ LSeg(p11,p2)) \/ (L4 /\ LSeg(p11,p2)) by XBOOLE_1:23
    .= {p11} by A90,A91,TOPREAL1:24,ZFMISC_1:39;
    then A92: L3 \/ L4 \/ LSeg(p11,p2) is_an_arc_of p00,p2 by A88,TOPREAL1:16;
      LSeg(p1,p00) /\ LSeg(p11,p2) c= {p01} by A7,A89,TOPREAL1:21,XBOOLE_1:27;
    then A93: LSeg(p1,p00) /\ LSeg(p11,p2) = {p01} or
     LSeg(p1,p00) /\ LSeg(p11,p2) = {} by ZFMISC_1:39;
A94:    now assume p01 in LSeg(p1,p00) /\ LSeg(p11,p2);
     then p01 in LSeg(p00,p1) & p01 in LSeg(p2,p11) & p00`2 <= p1`2 &
      p2`1 <= p11`1 by A5,A6,A86,EUCLID:56,XBOOLE_0:def 3;
     then p01`2 <= p1`2 & p2`1 <= p01`1 by TOPREAL1:9,10;
     then A95: p01`2 = p1`2 & p01`2 = p2`2 & p01`1 = p1`1 & p01`1 = p2`1
      by A5,A6,A86,Lm4,AXIOMS:21;
     then p1 = |[p01`1,p01`2]| by EUCLID:57 .= p2 by A95,EUCLID:57;
     hence contradiction by A1;
    end;
      LSeg(p1,p00) /\ (L3 \/ L4 \/ LSeg(p11,p2))
      = (LSeg(p1,p00) /\ (L3 \/ L4)) \/ (LSeg(p1,p00) /\
 LSeg(p11,p2)) by XBOOLE_1:23
     .= (LSeg(p1,p00) /\ L3) \/ (LSeg(p1,p00) /\ L4) by A93,A94,XBOOLE_1:23,
ZFMISC_1:37
     .= {p00} by A10,A11,A12,XBOOLE_0:def 10;
    hence P2 is_an_arc_of p1,p2 by A92,TOPREAL1:17;
   A96: L1 = LSeg(p1,p01) \/ LSeg(p1,p00) by A3,TOPREAL1:11;
   A97: L2 = LSeg(p11,p2) \/ LSeg(p01,p2) by A85,TOPREAL1:11;
    thus P1 \/ P2
     = LSeg(p01,p2) \/ (LSeg(p1,p01) \/ (LSeg(p1,p00) \/
        (L3 \/ L4 \/ LSeg(p11,p2)))) by XBOOLE_1:4
    .= L1 \/ (L3 \/ L4 \/ LSeg(p11,p2)) \/ LSeg(p01,p2) by A96,XBOOLE_1:4
    .= L1 \/ ((L3 \/ L4 \/ LSeg(p11,p2)) \/ LSeg(p01,p2)) by XBOOLE_1:4
    .= L1 \/ (L2 \/ (L3 \/ L4)) by A97,XBOOLE_1:4
    .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
   A98: LSeg(p01,p2) c= L2 by A85,Lm16,TOPREAL1:12;
   A99: {p1} = LSeg(p1,p01) /\ LSeg(p1,p00) by A3,TOPREAL1:14;
   A100: LSeg(p01,p2) /\ LSeg(p11,p2) = {p2} by A85,TOPREAL1:14;
         LSeg(p01,p2) /\ L3 c= L2 /\ L3 by A98,XBOOLE_1:27;
   then A101: LSeg(p01,p2) /\ L3 = {} by Lm2,XBOOLE_1:3;
   A102:  P1 /\ P2 = (LSeg(p1,p01) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/
 LSeg(p11,p2))))
         \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2))))
         by XBOOLE_1:23
     .= (LSeg(p1,p01) /\ LSeg(p1,p00))
        \/ (LSeg(p1,p01) /\ (L3 \/ L4 \/ LSeg(p11,p2)))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ (L3 \/ L4)) \/ (LSeg(p1,p01) /\
 LSeg(p11,p2)))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2))))
         by A99,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4)
        \/ (LSeg(p1,p01) /\ LSeg(p11,p2)))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4)
        \/ (LSeg(p1,p01) /\ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p00))
        \/ (LSeg(p01,p2) /\ (L3 \/ L4 \/ LSeg(p11,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4)
        \/ (LSeg(p1,p01) /\ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p00))
        \/ ((LSeg(p01,p2) /\ (L3 \/ L4)) \/ {p2})) by A100,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4)
        \/ (LSeg(p1,p01) /\ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p00))
        \/ ((LSeg(p01,p2) /\ L3) \/ (LSeg(p01,p2) /\ L4) \/ {p2})) by XBOOLE_1:
23
     .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p00)) \/ (LSeg(p01,p2) /\ L4 \/ {p2}))
         by A14,A101;
A103:    now per cases;
      suppose A104: p2 = p11;
       then A105: not p2 in LSeg(p1,p01) by A8,Lm4,TOPREAL1:9;
         LSeg(p1,p01) /\ LSeg(p11,p2) = LSeg(p1,p01) /\ {p2} by A104,TOPREAL1:7
             .= {} by A105,Lm1;
       hence P1 /\ P2
        = {p1} \/ (LSeg(p1,p01) /\ L3) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00))
           \/ {p2}) by A102,A104,TOPREAL1:24;
      suppose A106: p2 <> p11 & p2 <> p01;
        A107: LSeg(p01,p2) /\ L4 c= {p11} by A98,TOPREAL1:24,XBOOLE_1:27;
           now assume p11 in LSeg(p01,p2) /\ L4;
          then p11 in
 LSeg(p01,p2) & p01`1 <= p2`1 by A86,EUCLID:56,XBOOLE_0:def 3;
          then p11`1 <= p2`1 by TOPREAL1:9;
          then p2`1 = p11`1 & p2`2 = p11`2 by A86,Lm4,AXIOMS:21;
          hence contradiction by A106,Lm4,EUCLID:57;
         end;
        then {p11} <> LSeg(p01,p2) /\ L4 by ZFMISC_1:37;
       then A108: LSeg(p01,p2) /\ L4 = {} by A107,ZFMISC_1:39;
      A109: LSeg(p1,p01) /\
 LSeg(p11,p2) c= {p01} by A8,A89,TOPREAL1:21,XBOOLE_1:27;
          now assume p01 in LSeg(p1,p01) /\ LSeg(p11,p2);
         then p01 in LSeg(p2,p11) & p2`1 <= p11`1 by A86,EUCLID:56,XBOOLE_0:def
3;
         then p2`1 = 0 & p2`2 = 1 by A86,Lm4,TOPREAL1:9;
         hence contradiction by A106,EUCLID:57;
        end;
        then {p01} <> LSeg(p1,p01) /\ LSeg(p11,p2) by ZFMISC_1:37;
       then LSeg(p1,p01) /\ LSeg(p11,p2) = {} by A109,ZFMISC_1:39;
       hence P1 /\ P2
         = {p1} \/ (LSeg(p1,p01) /\ L3) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00)) \/
          {p2}) by A102,A108;
      suppose A110: p2 = p01;
       then A111: LSeg(p01,p2) /\ L4 = {p01} /\ L4 by TOPREAL1:7
            .= {} by Lm1,Lm8;
        A112: LSeg(p1,p01) /\
 LSeg(p11,p2) c= {p2} by A8,A110,TOPREAL1:21,XBOOLE_1:27;
             p2 in LSeg(p1,p01) by A110,TOPREAL1:6;
           then LSeg(p1,p01) /\ LSeg(p11,p2) <> {}
 by A110,Lm16,XBOOLE_0:def 3;
       then LSeg(p1,p01) /\ LSeg(p11,p2) = {p2} by A112,ZFMISC_1:39;
       hence P1 /\ P2
         = ({p1} \/ (LSeg(p1,p01) /\ L3)) \/ {p2} \/ ((LSeg(p01,p2) /\
 LSeg(p1,p00))
             \/ {p2}) by A102,A111,XBOOLE_1:4
        .= ({p1} \/ (LSeg(p1,p01) /\ L3)) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00))
             \/ {p2} \/ {p2}) by XBOOLE_1:4
        .= ({p1} \/ (LSeg(p1,p01) /\ L3)) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00))
             \/ ({p2} \/ {p2})) by XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p01) /\ L3) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00))
             \/ {p2});
        end;
         now per cases;
        suppose A113: p1 = p01;
         then A114: LSeg(p1,p01) /\ L3 = {p1} /\ L3 by TOPREAL1:7 .= {}
           by A113,Lm1,Lm7;
               LSeg(p01,p2) /\ LSeg(p1,p00) c= L2 /\ L1 & p1 in LSeg(p01,p2)
              by A98,A113,TOPREAL1:6,XBOOLE_1:27;
             then LSeg(p01,p2) /\ LSeg(p1,p00) c= {p1} &
              LSeg(p01,p2) /\ LSeg(p1,p00) <> {}
               by A113,Lm15,TOPREAL1:21,XBOOLE_0:def 3;
        then LSeg(p01,p2) /\ LSeg(p1,p00) = {p1} by ZFMISC_1:39;
        hence P1 /\ P2
           = {p1} \/ {p1} \/ {p2} by A103,A114,XBOOLE_1:4
          .= {p1,p2} by ENUMSET1:41;
        suppose A115: p1 = p00;
         A116: not p00 in LSeg(p01,p2) by A98,Lm4,TOPREAL1:10;
               LSeg(p01,p2) /\ LSeg(p1,p00) = LSeg(p01,p2) /\ {p00}
              by A115,TOPREAL1:7
              .= {} by A116,Lm1;
        hence P1 /\ P2
          = {p1,p2} by A103,A115,ENUMSET1:41,TOPREAL1:23;
        suppose A117: p1 <> p00 & p1 <> p01;
          A118: LSeg(p1,p01) /\ L3 c= {p00} by A8,TOPREAL1:23,XBOOLE_1:27;
               now assume p00 in LSeg(p1,p01) /\ L3;
              then p00 in LSeg(p1,p01) & p1`2 <= p01`2
               by A5,A6,EUCLID:56,XBOOLE_0:def 3;
              then p1`2 = 0 by A5,A6,Lm4,TOPREAL1:10;
              hence contradiction by A5,A6,A117,EUCLID:57;
             end;
             then {p00} <> LSeg(p1,p01) /\ L3 by ZFMISC_1:37;
         then A119: LSeg(p1,p01) /\ L3 = {} by A118,ZFMISC_1:39;
             A120: LSeg(p01,p2) /\ LSeg(p1,p00) c= {p01} by A7,A98,TOPREAL1:21,
XBOOLE_1:27;
               now assume p01 in LSeg(p01,p2) /\ LSeg(p1,p00);
              then p01 in LSeg(p00,p1) & p00`2 <= p1`2
               by A5,A6,EUCLID:56,XBOOLE_0:def 3;
              then p01`2 <= p1`2 by TOPREAL1:10;
              then p1`2 = 1 by A5,A6,Lm4,AXIOMS:21;
              hence contradiction by A5,A6,A117,EUCLID:57;
             end;
             then {p01} <> LSeg(p01,p2) /\ LSeg(p1,p00) by ZFMISC_1:37;
         then LSeg(p01,p2) /\ LSeg(p1,p00) = {} by A120,ZFMISC_1:39;
        hence P1 /\ P2
          = {p1,p2} by A103,A119,ENUMSET1:41;
       end;
      hence P1 /\ P2 = {p1,p2};
    suppose A121: p2 in L3;
     then A122: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 0 by TOPREAL1:19;
    take P1 = LSeg(p1,p00) \/ LSeg(p00,p2),P2
 = LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2));
      p00 in LSeg(p1,p00) & p00 in LSeg(p00,p2) by TOPREAL1:6;
    then LSeg(p1,p00) c= L1 & LSeg(p00,p2) c= L3 &
     p00 in LSeg(p1,p00) /\ LSeg(p00,p2) by A3,A121,Lm13,Lm14,TOPREAL1:12,
XBOOLE_0:def 3;
    then LSeg(p1,p00) /\ LSeg(p00,p2) c= L1 /\ L3 & L1 /\ L3 = {p00}
     & LSeg(p1,p00) /\ LSeg(p00,p2) <> {} by TOPREAL1:23,XBOOLE_1:27;
    then LSeg(p1,p00) /\ LSeg(p00,p2) = {p00} & (p1 <> p00 or p00 <> p2)
     by A1,ZFMISC_1:39;
    hence P1 is_an_arc_of p1,p2 by TOPREAL1:18;
      L2 is_an_arc_of p01,p11 & L4 is_an_arc_of p11,p10
     by Lm4,TOPREAL1:15;
    then A123: L2 \/ L4 is_an_arc_of p01,p10 by TOPREAL1:5,24;
A124: LSeg(p10,p2) c= L3 by A121,Lm17,TOPREAL1:12;
    then L2 /\ LSeg(p10,p2) c= L2 /\ L3 by XBOOLE_1:26;
then A125: L2 /\ LSeg(p10,p2) = {} by Lm2,XBOOLE_1:3;
      p10 in LSeg(p10,p2) by TOPREAL1:6;
    then A126: L4 /\ LSeg(p10,p2) c= L4 /\ L3 & p10 in L4 /\ LSeg(p10,p2)
     by A124,Lm18,XBOOLE_0:def 3,XBOOLE_1:27;
      (L2 \/ L4) /\ LSeg(p10,p2)
     = (L2 /\ LSeg(p10,p2)) \/ (L4 /\ LSeg(p10,p2)) by XBOOLE_1:23
    .= {p10} by A125,A126,TOPREAL1:22,ZFMISC_1:39;
    then A127: L2 \/ L4 \/
 LSeg(p10,p2) is_an_arc_of p01,p2 by A123,TOPREAL1:16;
      LSeg(p1,p01) /\ LSeg(p10,p2) c= {p00} by A8,A124,TOPREAL1:23,XBOOLE_1:27;
    then A128: LSeg(p1,p01) /\ LSeg(p10,p2) = {p00} or
     LSeg(p1,p01) /\ LSeg(p10,p2) = {} by ZFMISC_1:39;
A129:    now assume p00 in LSeg(p1,p01) /\ LSeg(p10,p2);
     then p00 in LSeg(p1,p01) & p00 in LSeg(p2,p10) & p1`2 <= p01`2 &
      p2`1 <= p10`1 by A5,A6,A122,EUCLID:56,XBOOLE_0:def 3;
     then A130: p00`2 = p1`2 & p00`2 = p2`2 & p00`1 = p1`1 & p00`1 = p2`1
      by A5,A6,A122,Lm4,TOPREAL1:9,10;
     then p1 = |[p00`1,p00`2]| by EUCLID:57 .= p2 by A130,EUCLID:57;
     hence contradiction by A1;
    end;
      LSeg(p1,p01) /\ (L2 \/ L4 \/ LSeg(p10,p2))
      = (LSeg(p1,p01) /\ (L2 \/ L4)) \/ (LSeg(p1,p01) /\
 LSeg(p10,p2)) by XBOOLE_1:23
     .= (LSeg(p1,p01) /\ L2) \/ (LSeg(p01,p1) /\ L4) by A128,A129,XBOOLE_1:23,
ZFMISC_1:37
     .= {p01} by A14,A15,A16,XBOOLE_0:def 10;
    hence P2 is_an_arc_of p1,p2 by A127,TOPREAL1:17;
   A131: L1 = LSeg(p1,p00) \/ LSeg(p1,p01) by A3,TOPREAL1:11;
   A132: L3 = LSeg(p10,p2) \/ LSeg(p00,p2) by A121,TOPREAL1:11;
    thus P1 \/ P2
     = LSeg(p00,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p01) \/
        (L2 \/ L4 \/ LSeg(p10,p2)))) by XBOOLE_1:4
    .= L1 \/ (L2 \/ L4 \/ LSeg(p10,p2)) \/ LSeg(p00,p2) by A131,XBOOLE_1:4
    .= L1 \/ ((L2 \/ L4 \/ LSeg(p10,p2)) \/ LSeg(p00,p2)) by XBOOLE_1:4
    .= L1 \/ (L2 \/ L4 \/ (LSeg(p10,p2) \/ LSeg(p00,p2))) by XBOOLE_1:4
    .= L1 \/ (L2 \/ (L3 \/ L4)) by A132,XBOOLE_1:4
    .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
   A133: LSeg(p00,p2) c= L3 by A121,Lm14,TOPREAL1:12;
   A134: {p1} = LSeg(p1,p00) /\ LSeg(p1,p01) by A3,TOPREAL1:14;
   A135: LSeg(p00,p2) /\ LSeg(p10,p2) = {p2} by A121,TOPREAL1:14;
      LSeg(p00,p2) /\ L2 c= L3 /\ L2 by A133,XBOOLE_1:27;
   then A136: LSeg(p00,p2) /\ L2 = {} by Lm2,XBOOLE_1:3;
   A137:  P1 /\ P2 = (LSeg(p1,p00) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/
 LSeg(p10,p2))))
         \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2))))
         by XBOOLE_1:23
     .= (LSeg(p1,p00) /\ LSeg(p1,p01))
        \/ (LSeg(p1,p00) /\ (L2 \/ L4 \/ LSeg(p10,p2)))
        \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ (L2 \/ L4)) \/ (LSeg(p1,p00) /\
 LSeg(p10,p2)))
        \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2))))
         by A134,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4)
        \/ (LSeg(p1,p00) /\ LSeg(p10,p2)))
        \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4)
        \/ (LSeg(p1,p00) /\ LSeg(p10,p2)))
        \/ ((LSeg(p00,p2) /\ LSeg(p1,p01))
        \/ (LSeg(p00,p2) /\ (L2 \/ L4 \/ LSeg(p10,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4)
        \/ (LSeg(p1,p00) /\ LSeg(p10,p2)))
        \/ ((LSeg(p00,p2) /\ LSeg(p1,p01))
        \/ ((LSeg(p00,p2) /\ (L2 \/ L4)) \/ {p2})) by A135,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4)
        \/ (LSeg(p1,p00) /\ LSeg(p10,p2)))
        \/ ((LSeg(p00,p2) /\ LSeg(p1,p01))
        \/ ((LSeg(p00,p2) /\ L2) \/ (LSeg(p00,p2) /\ L4) \/ {p2})) by XBOOLE_1:
23
     .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ LSeg(p10,p2)))
        \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/ (LSeg(p00,p2) /\ L4 \/ {p2}))
         by A10,A136;
A138:   now per cases;
      suppose A139: p2 = p10;
        then not p2 in LSeg(p1,p00) by A7,Lm4,TOPREAL1:9;
       then A140: LSeg(p1,p00) misses {p2} by ZFMISC_1:56;
         LSeg(p1,p00) /\ LSeg(p10,p2) = LSeg(p1,p00) /\ {p2} by A139,TOPREAL1:7
             .= {} by A140,XBOOLE_0:def 7;
      hence P1 /\ P2
        = {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01))
           \/ {p2}) by A137,A139,TOPREAL1:22;
      suppose A141: p2 <> p10 & p2 <> p00;
        A142: LSeg(p00,p2) /\ L4 c= {p10} by A133,TOPREAL1:22,XBOOLE_1:27;
           now assume p10 in LSeg(p00,p2) /\ L4;
          then p10 in LSeg(p00,p2) & p00`1 <=
 p2`1 by A122,EUCLID:56,XBOOLE_0:def 3;
          then p10`1 <= p2`1 by TOPREAL1:9;
          then p2`1 = p10`1 & p2`2 = p10`2 by A122,Lm4,AXIOMS:21;
          hence contradiction by A141,Lm4,EUCLID:57;
         end;
        then {p10} <> LSeg(p00,p2) /\ L4 by ZFMISC_1:37;
       then A143: LSeg(p00,p2) /\ L4 = {} by A142,ZFMISC_1:39;
      A144: LSeg(p1,p00) /\ LSeg(p10,p2) c= {p00} by A7,A124,TOPREAL1:23,
XBOOLE_1:27;
          now assume p00 in LSeg(p1,p00) /\ LSeg(p10,p2);
         then p00 in
 LSeg(p2,p10) & p2`1 <= p10`1 by A122,EUCLID:56,XBOOLE_0:def 3;
         then p2`1 = 0 & p2`2 = 0 by A122,Lm4,TOPREAL1:9;
         hence contradiction by A141,EUCLID:57;
        end;
        then {p00} <> LSeg(p1,p00) /\ LSeg(p10,p2) by ZFMISC_1:37;
       then LSeg(p1,p00) /\ LSeg(p10,p2) = {} by A144,ZFMISC_1:39;
       hence P1 /\ P2
         = {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/
 {p2})
            by A137,A143;
      suppose A145: p2 = p00;
      then A146: LSeg(p00,p2) /\ L4 = {p00} /\ L4 by TOPREAL1:7
                           .= {} by Lm1,Lm5;
        A147: LSeg(p1,p00) /\
 LSeg(p10,p2) c= {p2} by A7,A145,TOPREAL1:23,XBOOLE_1:27;
             p2 in LSeg(p1,p00) by A145,TOPREAL1:6;
           then LSeg(p1,p00) /\ LSeg(p10,p2) <> {}
 by A145,Lm14,XBOOLE_0:def 3;
       then LSeg(p1,p00) /\ LSeg(p10,p2) = {p2} by A147,ZFMISC_1:39;
       hence P1 /\ P2
         = ({p1} \/ (LSeg(p1,p00) /\ L2)) \/ {p2} \/ ((LSeg(p00,p2) /\
 LSeg(p1,p01))
             \/ {p2}) by A137,A146,XBOOLE_1:4
        .= ({p1} \/ (LSeg(p1,p00) /\ L2)) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01))
             \/ {p2} \/ {p2}) by XBOOLE_1:4
        .= ({p1} \/ (LSeg(p1,p00) /\ L2)) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01))
             \/ ({p2} \/ {p2})) by XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/
 {p2});
      end;
         now per cases;
        suppose A148: p1 = p01;
         then A149: LSeg(p00,p2) /\ LSeg(p1,p01) = LSeg(p00,p2) /\ {p1} by
TOPREAL1:7;
            not p1 in LSeg(p00,p2) by A133,A148,Lm4,TOPREAL1:10;
          then LSeg(p00,p2) /\ LSeg(p1,p01) = {} by A149,Lm1;
        hence P1 /\ P2
           = {p1,p2} by A138,A148,ENUMSET1:41,TOPREAL1:21;
        suppose A150: p1 = p00;
           then LSeg(p1,p00) /\ L2 = {p1} /\ L2 by TOPREAL1:7;
         then A151:LSeg(p1,p00) /\ L2 = {} by A150,Lm1,Lm6;
A152:     LSeg(p00,p2) /\ LSeg(p1,p01) c= L3 /\ L1 by A133,A150,XBOOLE_1:26;
               p00 in LSeg(p00,p2) by TOPREAL1:6;
             then LSeg(p00,p2) /\ LSeg(p1,p01) <> {} by A150,Lm13,XBOOLE_0:def
3
;
         then LSeg(p00,p2) /\ LSeg(p1,p01) = {p1} by A150,A152,TOPREAL1:23,
ZFMISC_1:39;
        hence P1 /\ P2
          = {p1} \/ {p1} \/ {p2} by A138,A151,XBOOLE_1:4
         .= {p1,p2} by ENUMSET1:41;
        suppose A153: p1 <> p00 & p1 <> p01;
          A154: LSeg(p1,p00) /\ L2 c= {p01} by A7,TOPREAL1:21,XBOOLE_1:27;
               now assume p01 in LSeg(p1,p00) /\ L2;
              then p01 in LSeg(p00,p1) & p00`2 <= p1`2
               by A5,A6,EUCLID:56,XBOOLE_0:def 3;
              then p01`2 <= p1`2 by TOPREAL1:10;
              then p1`2 = 1 by A5,A6,Lm4,AXIOMS:21;
              hence contradiction by A5,A6,A153,EUCLID:57;
             end;
             then {p01} <> LSeg(p1,p00) /\ L2 by ZFMISC_1:37;
         then A155: LSeg(p1,p00) /\ L2 = {} by A154,ZFMISC_1:39;
A156:         LSeg(p00,p2) /\ LSeg(p1,p01) c= L3 /\ L1 by A8,A133,XBOOLE_1:27;
               now assume p00 in LSeg(p00,p2) /\ LSeg(p1,p01);
              then p00 in LSeg(p1,p01) & p1`2 <= p01`2
               by A5,A6,EUCLID:56,XBOOLE_0:def 3;
              then p1`2 = 0 by A5,A6,Lm4,TOPREAL1:10;
              hence contradiction by A5,A6,A153,EUCLID:57;
             end;
             then {p00} <> LSeg(p00,p2) /\ LSeg(p1,p01) by ZFMISC_1:37;
         then LSeg(p00,p2) /\ LSeg(p1,p01) = {}
 by A156,TOPREAL1:23,ZFMISC_1:39;
        hence P1 /\ P2
          = {p1,p2} by A138,A155,ENUMSET1:41;
       end;
     hence P1 /\ P2 = {p1,p2};
    suppose A157: p2 in L4;
     then A158: ex q st q = p2 & q`1 = 1 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:19;
     take P1 = LSeg(p1,p00) \/ L3 \/ LSeg(p10,p2),P2
 = LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2);
       LSeg(p10,p2) c= L4 & p10 in L3 & p10 in LSeg(p10,p2)
      by A157,Lm18,TOPREAL1:6,12;
     then L3 /\ LSeg(p10,p2) c= {p10} & L3 /\ LSeg(p10,p2) <> {}
      by TOPREAL1:22,XBOOLE_0:def 3,XBOOLE_1:27;
     then L3 /\ LSeg(p10,p2) = {p10} by ZFMISC_1:39;
     then A159: L3 \/ LSeg(p10,p2) is_an_arc_of p00,p2 by Lm4,TOPREAL1:18;
       LSeg(p10,p2) c= L4 by A157,Lm18,TOPREAL1:12;
     then LSeg(p1,p00) /\ LSeg(p10,p2) c= L1 /\ L4 by A7,XBOOLE_1:27;
  then A160: LSeg(p1,p00) /\ LSeg(p10,p2) = {} by Lm3,XBOOLE_1:3;
       LSeg(p1,p00) /\ (L3 \/ LSeg(p10,p2))
       = (LSeg(p1,p00) /\ L3) \/ (LSeg(p1,p00) /\ LSeg(p10,p2)) by XBOOLE_1:23
      .= {p00} by A11,A12,A160,XBOOLE_0:def 10;
     then LSeg(p1,p00) \/ (L3 \/ LSeg(p10,p2)) is_an_arc_of p1,p2
      by A159,TOPREAL1:17;
     hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4;
 A161: LSeg(p11,p2) c= L4 by A157,Lm20,TOPREAL1:12;
     then L2 /\ LSeg(p11,p2) c= L2 /\ L4 & p11 in LSeg(p11,p2)
      by TOPREAL1:6,XBOOLE_1:27;
     then L2 /\ LSeg(p11,p2) c= {p11} & L2 /\ LSeg(p11,p2) <> {}
      by Lm19,TOPREAL1:24,XBOOLE_0:def 3;
     then L2 /\ LSeg(p11,p2) = {p11} by ZFMISC_1:39;
     then A162: L2 \/ LSeg(p11,p2) is_an_arc_of p01,p2 by Lm4,TOPREAL1:18;
       LSeg(p1,p01) /\ LSeg(p11,p2) c= L1 /\ L4 by A8,A161,XBOOLE_1:27;
 then A163: LSeg(p1,p01) /\ LSeg(p11,p2) = {} by Lm3,XBOOLE_1:3;
       LSeg(p1,p01) /\ (L2 \/ LSeg(p11,p2))
       = (LSeg(p1,p01) /\ L2) \/ (LSeg(p1,p01) /\ LSeg(p11,p2)) by XBOOLE_1:23
      .= {p01} by A15,A16,A163,XBOOLE_0:def 10;
     then LSeg(p1,p01) \/ (L2 \/ LSeg(p11,p2)) is_an_arc_of p1,p2
      by A162,TOPREAL1:17;
     hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4;
     thus R^2-unit_square = LSeg(p1,p00) \/ LSeg(p1,p01) \/ L2 \/ (L3 \/
 L4) by A3,TOPREAL1:11,20
      .= LSeg(p1,p00) \/ (LSeg(p1,p01) \/ L2) \/ (L3 \/ L4) by XBOOLE_1:4
      .= LSeg(p1,p00) \/ ((LSeg(p1,p01) \/ L2) \/ (L3 \/ L4)) by XBOOLE_1:4
      .= LSeg(p1,p00) \/ (L3 \/ (LSeg(p1,p01) \/ L2 \/ L4)) by XBOOLE_1:4
      .= LSeg(p1,p00) \/ L3 \/ (LSeg(p1,p01) \/ L2 \/ L4) by XBOOLE_1:4
      .= LSeg(p1,p00) \/ L3 \/ (LSeg(p1,p01) \/ L2
          \/ (LSeg(p11,p2) \/ LSeg(p10,p2))) by A157,TOPREAL1:11
      .= LSeg(p1,p00) \/ L3 \/ (LSeg(p10,p2)
          \/ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:4
      .= P1 \/ P2 by XBOOLE_1:4;
          p1 in LSeg(p1,p00) & p1 in LSeg(p1,p01) by TOPREAL1:6;
        then p1 in LSeg(p1,p00) /\ LSeg(p1,p01) by XBOOLE_0:def 3;
    then A164:  {p1} c= LSeg(p1,p00) /\ LSeg(p1,p01) by ZFMISC_1:37;
          now let a; assume A165: a in LSeg(p1,p00) /\ LSeg(p1,p01);
         then A166: a in LSeg(p00,p1) & a in LSeg(p1,p01) by XBOOLE_0:def 3;
         reconsider p = a as Point of TOP-REAL 2 by A165;
           p`1 <= p1`1 & p1`1 <= p`1 & p`2 <= p1`2 & p1`2 <= p`2
          by A5,A6,A166,Lm4,TOPREAL1:9,10;
         then p`1 = p1`1 & p`2 = p1`2 by AXIOMS:21;
         then a = |[p1`1,p1`2]| by EUCLID:57 .= p1 by EUCLID:57;
         hence a in {p1} by TARSKI:def 1;
        end;
        then LSeg(p1,p00) /\ LSeg(p1,p01) c= {p1} by TARSKI:def 3;
    then A167: LSeg(p1,p00) /\ LSeg(p1,p01) = {p1} by A164,XBOOLE_0:def 10;
          LSeg(p1,p00) /\ LSeg(p11,p2) c= L1 /\ L4 by A7,A161,XBOOLE_1:27;
    then A168: LSeg(p1,p00) /\ LSeg(p11,p2) = {} by Lm3,XBOOLE_1:3;
          p2 in LSeg(p10,p2) & p2 in LSeg(p11,p2) by TOPREAL1:6;
        then p2 in LSeg(p10,p2) /\ LSeg(p11,p2) by XBOOLE_0:def 3;
    then A169:  {p2} c= LSeg(p10,p2) /\ LSeg(p11,p2) by ZFMISC_1:37;
           now let a; assume A170: a in LSeg(p10,p2) /\ LSeg(p11,p2);
          then A171: a in LSeg(p10,p2) & a in LSeg(p2,p11) by XBOOLE_0:def 3;
          reconsider p = a as Point of TOP-REAL 2 by A170;
            p`2 <= p2`2 & p2`2 <= p`2 & p`1 <= p2`1 & p2`1 <= p`1
           by A158,A171,Lm4,TOPREAL1:9,10;
          then p`2 = p2`2 & p`1 = p2`1 by AXIOMS:21;
          then a = |[p2`1,p2`2]| by EUCLID:57 .= p2 by EUCLID:57;
          hence a in {p2} by TARSKI:def 1;
         end;
then A172:        LSeg(p10,p2) /\ LSeg(p11,p2) c= {p2} by TARSKI:def 3;
    A173: LSeg(p10,p2) c= L4 by A157,Lm18,TOPREAL1:12;
        then LSeg(p10,p2) /\ LSeg(p1,p01) c= L4 /\ L1 by A8,XBOOLE_1:27;
    then A174: LSeg(p10,p2) /\ LSeg(p1,p01) = {} by Lm3,XBOOLE_1:3;
A175:  P1 /\ P2
      = (LSeg(p1,p00) \/ L3) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:
23
     .= (LSeg(p1,p00) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2)))
         \/ (L3 /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:
23
     .= (LSeg(p1,p00) /\ (LSeg(p1,p01) \/ L2))
         \/ (LSeg(p1,p00) /\ LSeg(p11,p2))
         \/ (L3 /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:
23
     .= (LSeg(p1,p00) /\ LSeg(p1,p01)) \/ (LSeg(p1,p00) /\ L2)
         \/ (L3 /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/
 LSeg(p11,p2))) by A168,XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p00) /\ L2)
         \/ ((L3 /\ (LSeg(p1,p01) \/ L2)) \/ (L3 /\ LSeg(p11,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/
 LSeg(p11,p2))) by A167,XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p00) /\ L2)
         \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ L2) \/ (L3 /\ LSeg(p11,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:
23
     .= {p1} \/ (LSeg(p1,p00) /\ L2)
         \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2)))
         \/ ((LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2))
         \/ (LSeg(p10,p2) /\ LSeg(p11,p2))) by Lm2,XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p00) /\ L2)
         \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2)))
         \/ ((LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2)) \/
 {p2}) by A169,A172,XBOOLE_0:def 10
     .= {p1} \/ (LSeg(p1,p00) /\ L2)
         \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2)))
         \/ ((LSeg(p10,p2) /\ LSeg(p1,p01)) \/ (LSeg(p10,p2) /\ L2) \/ {p2})
          by XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p00) /\ L2)
         \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2)))
         \/ ((LSeg(p10,p2) /\ L2) \/ {p2}) by A174;
A176:  now per cases;
     suppose A177: p2 = p11;
   then L3 /\ LSeg(p11,p2) = L3 /\ {p11} by TOPREAL1:7
                        .= {} by Lm1,Lm12;
     hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2) \/ (L3 /\ LSeg(p1,p01)) \/
 {p2}
           by A175,A177,TOPREAL1:24;
     suppose A178: p2 = p10;
    then LSeg(p10,p2) /\ L2 = {p10} /\ L2 by TOPREAL1:7
                     .= {} by Lm1,Lm10;
      hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2)
                 \/ ((L3 /\ LSeg(p1,p01)) \/ {p2} \/
 {p2}) by A175,A178,TOPREAL1:22,XBOOLE_1:4
             .= {p1} \/ (LSeg(p1,p00) /\ L2)
                 \/ ((L3 /\ LSeg(p1,p01)) \/ ({p2} \/ {p2})) by XBOOLE_1:4
             .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ (L3 /\ LSeg(p1,p01)) \/ {p2}
                 by XBOOLE_1:4;
     suppose A179: p2 <> p10 & p2 <> p11;
  A180:  L3 /\ LSeg(p11,p2) c= {p10} by A161,TOPREAL1:22,XBOOLE_1:27;
         now assume p10 in L3 /\ LSeg(p11,p2);
        then p10 in LSeg(p2,p11) & p2`2 <= p11`2 by A158,EUCLID:56,XBOOLE_0:def
3;
        then p2`2 = p10`2 & p2`1 = p10`1 by A158,Lm4,TOPREAL1:10;
        then p2 = |[p10`1,p10`2]| by EUCLID:57 .= p10 by EUCLID:57;
        hence contradiction by A179;
       end;
       then L3 /\ LSeg(p11,p2) <> {p10} by ZFMISC_1:37;
   then A181: L3 /\ LSeg(p11,p2) = {} by A180,ZFMISC_1:39;
A182:  LSeg(p10,p2) /\ L2 c= L4 /\ L2 by A173,XBOOLE_1:27;
         now assume p11 in LSeg(p10,p2) /\ L2;
        then p11 in LSeg(p10,p2) & p10`2 <= p2`2 by A158,EUCLID:56,XBOOLE_0:def
3;
        then p11`2 <= p2`2 by TOPREAL1:10;
        then p11`2 = p2`2 & p11`1 = p2`1 by A158,Lm4,AXIOMS:21;
        then p2 = |[p11`1,p11`2]| by EUCLID:57 .= p11 by EUCLID:57;
        hence contradiction by A179;
       end;
       then {p11} <> LSeg(p10,p2) /\ L2 by ZFMISC_1:37;
    then LSeg(p10,p2) /\ L2 = {} by A182,TOPREAL1:24,ZFMISC_1:39;
      hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2) \/ (L3 /\ LSeg(p1,p01)) \/
       {p2} by A175,A181;
     end;
        now per cases;
       suppose A183: p1 = p01;
   then L3 /\ LSeg(p1,p01) = L3 /\ {p01} by TOPREAL1:7
                        .= {} by Lm1,Lm7;
       hence P1 /\ P2 = {p1,p2} by A176,A183,ENUMSET1:41,TOPREAL1:21;
       suppose A184: p1 <> p01 & p1 <> p00;
A185:
       L3 /\ LSeg(p1,p01) c= L3 /\ L1 by A8,XBOOLE_1:27;
         now assume p00 in L3 /\ LSeg(p1,p01);
        then p00 in LSeg(p1,p01) by XBOOLE_0:def 3;
        then p1`2 = p00`2 & p1`1 = p00`1 by A5,A6,Lm4,TOPREAL1:10;
        then p1 = |[p00`1,p00`2]| by EUCLID:57 .= p00 by EUCLID:57;
        hence contradiction by A184;
       end;
       then {p00} <> L3 /\ LSeg(p1,p01) by ZFMISC_1:37;
   then A186: L3 /\ LSeg(p1,p01) = {} by A185,TOPREAL1:23,ZFMISC_1:39;
   A187:  LSeg(p1,p00) /\ L2 c= {p01} by A7,TOPREAL1:21,XBOOLE_1:27;
         now assume p01 in LSeg(p1,p00) /\ L2;
        then A188: p01 in LSeg(p00,p1) by XBOOLE_0:def 3;
          p00`2 <= p1`2 by A5,A6,EUCLID:56;
        then p01`2 <= p1`2 by A188,TOPREAL1:10;
        then p1`2 = p01`2 & p1`1 = p01`1 by A5,A6,Lm4,AXIOMS:21;
        then p1 = |[p01`1,p01`2]| by EUCLID:57 .= p01 by EUCLID:57;
        hence contradiction by A184;
       end;
       then {p01} <> LSeg(p1,p00) /\ L2 by ZFMISC_1:37;
       then LSeg(p1,p00) /\ L2 = {} by A187,ZFMISC_1:39;
       hence P1 /\ P2 = {p1,p2} by A176,A186,ENUMSET1:41;
       suppose A189: p1 = p00;
       then LSeg(p1,p00) /\ L2 = {p00} /\ L2 by TOPREAL1:7
                        .= {} by Lm1,Lm6;
       hence P1 /\ P2 = {p1,p2} by A176,A189,ENUMSET1:41,TOPREAL1:23;
      end;
    hence P1 /\ P2 = {p1,p2};
 end;

Lm24:
 p1 <> p2 & p2 in R^2-unit_square &
 p1 in LSeg(p01, p11) implies
  ex P1,P2 being non empty Subset of TOP-REAL 2
  st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
   R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2}
 proof assume that
 A1: p1 <> p2 and
 A2: p2 in R^2-unit_square and
 A3: p1 in LSeg(p01, p11);
A4:   p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:20,XBOOLE_0:def 2;
   consider q1 such that
  A5: q1 = p1 and A6: q1`1 <= 1 & q1`1 >= 0 & q1`2 = 1 by A3,TOPREAL1:19;
  A7: LSeg(p01,p1) c= L2 by A3,Lm16,TOPREAL1:12;
  A8: LSeg(p1,p11) c= L2 by A3,Lm19,TOPREAL1:12;
        LSeg(p1,p01) /\ L3 c= L2 /\ L3 by A7,XBOOLE_1:26;
  then A9: LSeg(p1,p01) /\ L3 = {} by Lm2,XBOOLE_1:3;
        p01 in LSeg(p01,p1) by TOPREAL1:6;
  then A10:  LSeg(p01,p1) /\ L1 <> {} by Lm15,XBOOLE_0:def 3;
      A11: LSeg(p01,p1) /\ L1 c= L2 /\ L1 by A7,XBOOLE_1:26;
        LSeg(p1,p11) /\ L3 c= L2 /\ L3 by A8,XBOOLE_1:26;
  then A12: LSeg(p1,p11) /\ L3 = {} by Lm2,XBOOLE_1:3;
        p11 in LSeg(p1,p11) by TOPREAL1:6;
   then A13: LSeg(p1,p11) /\ L4 <> {} by Lm20,XBOOLE_0:def 3;
      A14: LSeg(p1,p11) /\ L4 c= {p11} by A8,TOPREAL1:24,XBOOLE_1:26;
   per cases by A4,XBOOLE_0:def 2;
    suppose A15: p2 in L1;
     then A16: ex q2 st q2 = p2 & q2`1 = 0 & q2`2 <= 1 & q2`2 >= 0 by TOPREAL1:
19;
    A17: p1 <> p01 or p2 <> p01 by A1;
     take P1 = LSeg(p1,p01) \/ LSeg(p01,p2),P2
 = LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2));
       p01 in LSeg(p1,p01) & p01 in LSeg(p01,p2) by TOPREAL1:6;
     then LSeg(p1,p01) c= L2 & LSeg(p01,p2) c= L1 &
      p01 in LSeg(p1,p01) /\ LSeg(p01,p2) by A3,A15,Lm15,Lm16,TOPREAL1:12,
XBOOLE_0:def 3;
     then LSeg(p1,p01) /\ LSeg(p01,p2) c= L2 /\ L1
      & LSeg(p1,p01) /\ LSeg(p01,p2) <> {} by XBOOLE_1:27;
     then LSeg(p1,p01) /\ LSeg(p01,p2) = {p01} by TOPREAL1:21,ZFMISC_1:39;
     hence P1 is_an_arc_of p1,p2 by A17,TOPREAL1:18;
        L3 is_an_arc_of p10,p00 & L4 is_an_arc_of p11,p10 by Lm4,TOPREAL1:15;
     then A18: L3 \/ L4 is_an_arc_of p11,p00 by TOPREAL1:5,22;
 A19: LSeg(p00,p2) c= L1 & p00 in LSeg(p00,p2) by A15,Lm13,TOPREAL1:6,12;
     then A20: L3 /\ LSeg(p00,p2) c= {p00} & {} <> L3 /\ LSeg(p00,p2)
      by Lm14,TOPREAL1:23,XBOOLE_0:def 3,XBOOLE_1:26;
       L4 /\ LSeg(p00,p2) c= L4 /\ L1 by A19,XBOOLE_1:27;
 then A21: L4 /\ LSeg(p00,p2) = {} by Lm3,XBOOLE_1:3;
 A22: LSeg(p2,p01) c= L1 by A15,Lm15,TOPREAL1:12;
       (L3 \/ L4) /\ LSeg(p00,p2)
      = (L3 /\ LSeg(p00,p2)) \/ (L4 /\ LSeg(p00,p2)) by XBOOLE_1:23
     .= {p00} by A20,A21,ZFMISC_1:39;
     then A23: L3 \/ L4 \/ LSeg(p00,p2) is_an_arc_of p11,p2 by A18,TOPREAL1:16;
       LSeg(p1,p11) /\ LSeg(p00,p2) c= L2 /\ L1 by A8,A19,XBOOLE_1:27;
    then A24: LSeg(p1,p11) /\ LSeg(p00,p2) = {p01} or
     LSeg(p1,p11) /\ LSeg(p00,p2) = {} by TOPREAL1:21,ZFMISC_1:39;
A25:    now assume p01 in LSeg(p1,p11) /\ LSeg(p00,p2);
     then p01 in LSeg(p1,p11) & p01 in LSeg(p00,p2) & p1`1 <= p11`1 &
      p00`2 <= p2`2 by A5,A6,A16,EUCLID:56,XBOOLE_0:def 3;
     then p01`2 <= p2`2 & p1`1 <= p01`1 by TOPREAL1:9,10;
     then A26: p01`2 = p1`2 & p01`2 = p2`2 & p01`1 = p1`1 & p01`1 = p2`1
      by A5,A6,A16,Lm4,AXIOMS:21;
     then p1 = |[p01`1,p01`2]| by EUCLID:57 .= p2 by A26,EUCLID:57;
     hence contradiction by A1;
    end;
      LSeg(p1,p11) /\ (L3 \/ L4 \/ LSeg(p00,p2))
      = (LSeg(p1,p11) /\ (L3 \/ L4)) \/ (LSeg(p1,p11) /\
 LSeg(p00,p2)) by XBOOLE_1:23
     .= (LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ L4) by A24,A25,XBOOLE_1:23,
ZFMISC_1:37
     .= {p11} by A12,A13,A14,ZFMISC_1:39;
    hence P2 is_an_arc_of p1,p2 by A23,TOPREAL1:17;
A27: LSeg(p1,p01) \/ LSeg(p1,p11) = L2 by A3,TOPREAL1:11;
A28: LSeg(p00,p2) \/ LSeg(p01,p2) = L1 by A15,TOPREAL1:11;
    thus P1 \/ P2
     = LSeg(p01,p2) \/ (LSeg(p1,p01) \/ (LSeg(p1,p11) \/
        (L3 \/ L4 \/ LSeg(p00,p2)))) by XBOOLE_1:4
    .= LSeg(p01,p2) \/ (L2 \/ (L3 \/ L4 \/ LSeg(p00,p2))) by A27,XBOOLE_1:4
    .= LSeg(p01,p2) \/ (L2 \/ (L3 \/ L4) \/ LSeg(p00,p2)) by XBOOLE_1:4
    .= LSeg(p00,p2) \/ LSeg(p01,p2) \/ (L2 \/ (L3 \/ L4)) by XBOOLE_1:4
    .= R^2-unit_square by A28,TOPREAL1:20,XBOOLE_1:4;
   A29: {p1} = LSeg(p1,p01) /\ LSeg(p1,p11) by A3,TOPREAL1:14;
   A30: LSeg(p01,p2) /\ LSeg(p00,p2) = {p2} by A15,TOPREAL1:14;
         LSeg(p01,p2) /\ L4 c= {} by A22,Lm3,XBOOLE_1:27;
   then A31: LSeg(p01,p2) /\ L4 = {} by XBOOLE_1:3;
   A32:  P1 /\ P2 = (LSeg(p1,p01) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/
 LSeg(p00,p2))))
         \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2))))
         by XBOOLE_1:23
     .= (LSeg(p1,p01) /\ LSeg(p1,p11))
        \/ (LSeg(p1,p01) /\ (L3 \/ L4 \/ LSeg(p00,p2)))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ (L3 \/ L4)) \/ (LSeg(p1,p01) /\
 LSeg(p00,p2)))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2))))
         by A29,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4)
        \/ (LSeg(p1,p01) /\ LSeg(p00,p2)))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4)
        \/ (LSeg(p1,p01) /\ LSeg(p00,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p11))
        \/ (LSeg(p01,p2) /\ (L3 \/ L4 \/ LSeg(p00,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L4) \/ (LSeg(p1,p01) /\ LSeg(p00,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p11))
        \/ ((LSeg(p01,p2) /\ (L3 \/ L4)) \/ {p2})) by A9,A30,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p01) /\ L4) \/ (LSeg(p1,p01) /\ LSeg(p00,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p11))
        \/ ((LSeg(p01,p2) /\ L3) \/ (LSeg(p01,p2) /\ L4) \/ {p2})) by XBOOLE_1:
23
     .= {p1} \/ ((LSeg(p1,p01) /\ L4) \/ (LSeg(p1,p01) /\ LSeg(p00,p2)))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p11))
        \/ ((LSeg(p01,p2) /\ L3) \/ {p2})) by A31;
A33:  now per cases;
     suppose A34: p1 = p01;
    then LSeg(p1,p01) /\ L4 = {p1} /\ L4 by TOPREAL1:7;
 then A35: LSeg(p1,p01) /\ L4 = {} by A34,Lm1,Lm8;
  A36: LSeg(p01,p2) /\ LSeg(p1,p11) c= {p1} by A22,A34,TOPREAL1:21,XBOOLE_1:27;
       p1 in LSeg(p01,p2) by A34,TOPREAL1:6;
     then LSeg(p01,p2) /\ LSeg(p1,p11) <> {} by A34,Lm16,XBOOLE_0:def 3;
  then LSeg(p01,p2) /\ LSeg(p1,p11) = {p1} by A36,ZFMISC_1:39;
    hence P1 /\ P2 = {p1} \/ ({p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2)))
            \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A32,A35,XBOOLE_1:4
       .= {p1} \/ {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2))
            \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by XBOOLE_1:4
       .= {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2))
            \/ ((LSeg(p01,p2) /\ L3) \/ {p2});
     suppose A37: p1 = p11;
  then A38: LSeg(p01,p2) /\ LSeg(p1,p11) = LSeg(p01,p2) /\ {p1} by TOPREAL1:7;
        not p1 in LSeg(p01,p2) by A16,A37,Lm4,TOPREAL1:9;
  then LSeg(p01,p2) /\ LSeg(p1,p11) = {} by A38,Lm1;
     hence P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2))
        \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A32,A37,TOPREAL1:24,XBOOLE_1:4
      .= {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2))
        \/ ((LSeg(p01,p2) /\ L3) \/ {p2});
     suppose A39: p1 <> p11 & p1 <> p01;
  A40: LSeg(p1,p01) /\ L4 c= {p11} by A7,TOPREAL1:24,XBOOLE_1:27;
        now assume p11 in LSeg(p1,p01) /\ L4;
       then p11 in LSeg(p01,p1) & p01`1 <= p1`1 by A5,A6,EUCLID:56,XBOOLE_0:def
3;
       then p11`1 <= p1`1 by TOPREAL1:9;
       then p1`1 = 1 by A5,A6,Lm4,AXIOMS:21;
       hence contradiction by A5,A6,A39,EUCLID:57;
      end;
     then {p11} <> LSeg(p1,p01) /\ L4 by ZFMISC_1:37;
 then A41: LSeg(p1,p01) /\ L4 = {} by A40,ZFMISC_1:39;
  A42: LSeg(p01,p2) /\ LSeg(p1,p11) c= {p01} by A8,A22,TOPREAL1:21,XBOOLE_1:27;
       now assume p01 in LSeg(p01,p2) /\ LSeg(p1,p11);
      then p01 in LSeg(p1,p11) & p1`1 <= p11`1 by A5,A6,EUCLID:56,XBOOLE_0:def
3;
      then p1`1 = 0 by A5,A6,Lm4,TOPREAL1:9;
      hence contradiction by A5,A6,A39,EUCLID:57;
     end;
     then {p01} <> LSeg(p01,p2) /\ LSeg(p1,p11) by ZFMISC_1:37;
 then LSeg(p01,p2) /\ LSeg(p1,p11) = {} by A42,ZFMISC_1:39;
     hence P1 /\ P2 = {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2))
        \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A32,A41;
     end;
       now per cases;
     suppose A43: p2 <> p00 & p2 <> p01;
A44:      LSeg(p1,p01) /\ LSeg(p00,p2) c= L2 /\ L1 by A7,A19,XBOOLE_1:27;
        now assume p01 in LSeg(p1,p01) /\ LSeg(p00,p2);
       then p01 in LSeg(p00,p2) & p00`2 <= p2`2 by A16,EUCLID:56,XBOOLE_0:def 3
;
       then p01`2 <= p2`2 by TOPREAL1:10;
       then p2`2 = 1 by A16,Lm4,AXIOMS:21;
       hence contradiction by A16,A43,EUCLID:57;
      end;
      then {p01} <> LSeg(p1,p01) /\ LSeg(p00,p2) by ZFMISC_1:37;
   then A45: LSeg(p1,p01) /\ LSeg(p00,p2) = {} by A44,TOPREAL1:21,ZFMISC_1:39;
    A46: LSeg(p01,p2) /\ L3 c= {p00} by A22,TOPREAL1:23,XBOOLE_1:27;
          now assume p00 in LSeg(p01,p2) /\ L3;
         then p00 in LSeg(p2,p01) & p2`2 <= p01`2 by A16,EUCLID:56,XBOOLE_0:def
3;
         then 0 = p2`2 by A16,Lm4,TOPREAL1:10;
         hence contradiction by A16,A43,EUCLID:57;
        end;
       then {p00} <> LSeg(p01,p2) /\ L3 by ZFMISC_1:37;
   then LSeg(p01,p2) /\ L3 = {} by A46,ZFMISC_1:39;
       hence P1 /\ P2 = {p1,p2} by A33,A45,ENUMSET1:41;
       suppose A47:p2 = p00;
    then A48: LSeg(p1,p01) /\ LSeg(p00,p2) = LSeg(p1,p01) /\
 {p00} by TOPREAL1:7;
         not p00 in LSeg(p1,p01) by A7,Lm4,TOPREAL1:10;
        then LSeg(p1,p01) /\ LSeg(p00,p2) = {} by A48,Lm1;
       hence P1 /\ P2 = {p1,p2} by A33,A47,ENUMSET1:41,TOPREAL1:23;
       suppose A49: p2 = p01;
    then LSeg(p01,p2) /\ L3 = {p01} /\ L3 by TOPREAL1:7;
   then A50: LSeg(p01,p2) /\ L3 = {} by Lm1,Lm7;
A51:       LSeg(p1,p01) /\ LSeg(p00,p2) c= L2 /\ L1 by A7,A19,XBOOLE_1:27;
         p2 in LSeg(p1,p01) by A49,TOPREAL1:6;
       then {} <> LSeg(p1,p01) /\ LSeg(p00,p2) by A49,Lm15,XBOOLE_0:def 3;
   then LSeg(p1,p01) /\ LSeg(p00,p2) = {p2} by A49,A51,TOPREAL1:21,ZFMISC_1:39
;
       hence P1 /\ P2
          = {p1} \/ ({p2} \/ {p2}) by A33,A50,XBOOLE_1:4
         .= {p1,p2} by ENUMSET1:41;
      end;
    hence P1 /\ P2 = {p1,p2};
    suppose A52: p2 in L2;
     then consider q such that
 A53: q = p2 and A54: q`1 <= 1 & q`1 >= 0 & q`2 = 1 by TOPREAL1:19;
A55:     q1 = |[q1`1,q1`2]| & q = |[q`1,q`2]| by EUCLID:57;
 A56: LSeg(p2,p11) c= L2 by A52,Lm19,TOPREAL1:12;
 A57: LSeg(p2,p01) c= L2 by A52,Lm16,TOPREAL1:12;
 A58: LSeg(p1,p2) c= L2 by A3,A52,TOPREAL1:12;
        now per cases by A1,A5,A6,A53,A54,A55,REAL_1:def 5;
       suppose A59: q1`1 < q`1;
       take P1 = LSeg(p1,p2),P2 = LSeg(p1,p01) \/ (L1 \/ L3 \/ L4 \/
 LSeg(p11,p2));
       thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
 A60:    now assume
A61:      LSeg(p1,p01) /\ LSeg(p11,p2) <> {};
         consider a being Element of LSeg(p1,p01) /\ LSeg(p11,p2);
         reconsider p = a as Point of TOP-REAL 2 by A61,TARSKI:def 3;
           p in LSeg(p01,p1) & p in LSeg(p2,p11) & p01`1 <= p1`1 & p2`1 <= p11
`1
          by A5,A6,A53,A54,A61,EUCLID:56,XBOOLE_0:def 3;
         then p`1 <= p1`1 & p2`1 <= p`1 by TOPREAL1:9;
         hence contradiction by A5,A53,A59,AXIOMS:22;
        end;
  A62:   LSeg(p1,p01) /\ L4 c= {p11} by A7,TOPREAL1:24,XBOOLE_1:26;
          now assume p11 in LSeg(p1,p01) /\ L4;
         then p11 in LSeg(p01,p1) & p01`1 <=
 p1`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         then p11`1 <= p1`1 by TOPREAL1:9;
         hence contradiction by A5,A6,A54,A59,Lm4,AXIOMS:21;
        end;
       then {p11} <> LSeg(p1,p01) /\ L4 by ZFMISC_1:37;
 then A63:   LSeg(p1,p01) /\ L4 = {} by A62,ZFMISC_1:39;
         LSeg(p1,p01) /\ L3 c= L2 /\ L3 & L3 /\ L2 = {}
        by A7,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A64:   LSeg(p1,p01) /\ L3 = {} by XBOOLE_1:3;
A65:   LSeg(p1,p01) /\ L1 c= L2 /\ L1 by A7,XBOOLE_1:26;
         p01 in LSeg(p1,p01) by TOPREAL1:6;
       then A66: LSeg(p1,p01) /\ L1 <> {} by Lm15,XBOOLE_0:def 3;
 A67:   LSeg(p1,p01) /\ (L1 \/ L3 \/ L4 \/ LSeg(p11,p2))
         = LSeg(p1,p01) /\ (L1 \/ L3 \/ L4) \/ (LSeg(p1,p01) /\ LSeg(p11,p2))
            by XBOOLE_1:23
        .= LSeg(p1,p01) /\ (L1 \/ L3) \/ (LSeg(p1,p01) /\ L4) by A60,XBOOLE_1:
23
        .= LSeg(p1,p01) /\ L1 \/ (LSeg(p1,p01) /\ L3) by A63,XBOOLE_1:23
        .= {p01} by A64,A65,A66,TOPREAL1:21,ZFMISC_1:39;
      L1 is_an_arc_of p01,p00 by Lm4,TOPREAL1:15;
 then A68:   L1 \/ L3 is_an_arc_of p01,p10 by TOPREAL1:16,23;
         (L1 \/ L3) /\ L4 = L1 /\ L4 \/ L3 /\ L4 by XBOOLE_1:23
                     .= {p10} by Lm3,TOPREAL1:22;
 then A69:   L1 \/ L3 \/ L4 is_an_arc_of p01,p11 by A68,TOPREAL1:16;
 A70:  L4 /\ LSeg(p11,p2) c= L4 /\ L2 by A56,XBOOLE_1:26;
         p11 in LSeg(p11,p2) by TOPREAL1:6;
       then L4 /\ LSeg(p11,p2) <> {} by Lm20,XBOOLE_0:def 3;
 then A71:   L4 /\ LSeg(p11,p2) = {p11} by A70,TOPREAL1:24,ZFMISC_1:39;
  A72:   L1 /\ LSeg(p11,p2) c= {p01} by A56,TOPREAL1:21,XBOOLE_1:26;
          now assume p01 in L1 /\ LSeg(p11,p2);
         then p01 in
 LSeg(p2,p11) & p2`1 <= p11`1 by A53,A54,EUCLID:56,XBOOLE_0:def 3;
         hence contradiction by A6,A53,A59,Lm4,TOPREAL1:9;
        end;
       then {p01} <> L1 /\ LSeg(p11,p2) by ZFMISC_1:37;
 then A73:   L1 /\ LSeg(p11,p2) = {} by A72,ZFMISC_1:39;
         L3 /\ LSeg(p11,p2) c= L3 /\ L2 & L3 /\ L2 = {}
        by A56,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A74:   L3 /\ LSeg(p11,p2) = {} by XBOOLE_1:3;
         (L1 \/ L3 \/ L4) /\ LSeg(p11,p2)
         = (L1 \/ L3) /\ LSeg(p11,p2) \/ L4 /\ LSeg(p11,p2) by XBOOLE_1:23
        .= (L1 /\ LSeg(p11,p2)) \/ (L3 /\ LSeg(p11,p2)) \/
 {p11} by A71,XBOOLE_1:23
        .= {p11} by A73,A74;
       then L1 \/ L3 \/ L4 \/
 LSeg(p11,p2) is_an_arc_of p01,p2 by A69,TOPREAL1:16;
       hence P2 is_an_arc_of p1,p2 by A67,TOPREAL1:17;
       thus P1 \/ P2
         = LSeg(p01,p1) \/ LSeg(p1,p2) \/ (L1 \/ L3 \/ L4 \/
 LSeg(p11,p2)) by XBOOLE_1:4
        .= LSeg(p01,p1) \/ LSeg(p1,p2) \/ LSeg(p2,p11) \/ (L1 \/ L3 \/
 L4) by XBOOLE_1:4
        .= L2 \/ (L1 \/ L3 \/ L4) by A3,A52,TOPREAL1:13
        .= L2 \/ (L1 \/ (L3 \/ L4)) by XBOOLE_1:4
        .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
         p1 in LSeg(p1,p2) & p1 in LSeg(p1,p01) by TOPREAL1:6;
       then p1 in LSeg(p1,p2) /\ LSeg(p1,p01) by XBOOLE_0:def 3;
  then A75:   {p1} c= LSeg(p1,p2) /\ LSeg(p1,p01) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p1,p01) c= {p1}
        proof
         let a; assume A76: a in LSeg(p1,p2) /\ LSeg(p1,p01);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p1,p2) & p in LSeg(p01,p1) & p1`2 <= p2`2 & p1`1 <= p2`1
           & p01`1 <= p1`1 by A5,A6,A53,A54,A59,A76,EUCLID:56,XBOOLE_0:def 3;
          then p1`2 <= p`2 & p`2 <= p2`2 & p1`1 <= p`1 & p`1 <= p1`1
           by TOPREAL1:9,10;
          then p1`1 = p`1 & p`2 = 1 by A5,A6,A53,A54,AXIOMS:21;
          then p = |[p1`1, 1]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
          hence a in {p1} by TARSKI:def 1;
        end;
 then A77:   LSeg(p1,p2) /\ LSeg(p1,p01) = {p1} by A75,XBOOLE_0:def 10;
         p2 in LSeg(p1,p2) & p2 in LSeg(p11,p2) by TOPREAL1:6;
       then p2 in LSeg(p1,p2) /\ LSeg(p11,p2) by XBOOLE_0:def 3;
  then A78:   {p2} c= LSeg(p1,p2) /\ LSeg(p11,p2) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p11,p2) c= {p2}
        proof
         let a; assume A79: a in LSeg(p1,p2) /\ LSeg(p11,p2);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p1,p2) & p in LSeg(p2,p11) & p1`2 <= p2`2 & p1`1 <= p2`1
           & p2`1 <= p11`1 by A5,A6,A53,A54,A59,A79,EUCLID:56,XBOOLE_0:def 3;
          then p1`2 <= p`2 & p`2 <= p2`2 & p2`1 <= p`1 & p`1 <= p2`1
           by TOPREAL1:9,10;
          then p2`1 = p`1 & p`2 = 1 by A5,A6,A53,A54,AXIOMS:21;
          then p = |[ p2`1, 1]| by EUCLID:57 .= p2 by A53,A54,EUCLID:57;
          hence a in {p2} by TARSKI:def 1;
        end;
 then A80:   LSeg(p1,p2) /\ LSeg(p11,p2) = {p2} by A78,XBOOLE_0:def 10;
         LSeg(p1,p2) /\ L3 c= L2 /\ L3 & L3 /\ L2 = {}
        by A58,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A81:   LSeg(p1,p2) /\ L3 = {} by XBOOLE_1:3;
  A82:   P1 /\ P2
         = (LSeg(p1,p2) /\ LSeg(p1,p01))
           \/ (LSeg(p1,p2) /\ (L1 \/ L3 \/ L4 \/ LSeg(p11,p2))) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L3 \/ L4)) \/
 {p2}) by A77,A80,XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L3)) \/ (LSeg(p1,p2) /\ L4) \/ {p2})
            by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ (LSeg(p1,p2) /\ L3)
            \/ (LSeg(p1,p2) /\ L4) \/ {p2}) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2}))
            by A81,XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2})
            by XBOOLE_1:4;
A83:        LSeg(p1,p2) /\ L1 c= L2 /\ L1 by A58,XBOOLE_1:26;
  A84:    now per cases;
         suppose A85: p1 = p01;
            p1 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L1 <> {} by A85,Lm15,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L1 = {p1} by A83,A85,TOPREAL1:21,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A82;
         suppose A86: p1 <> p01;
             now assume p01 in LSeg(p1,p2) /\ L1;
            then p01 in LSeg(p1,p2) & p1`1 <= p2`1
             by A5,A53,A59,XBOOLE_0:def 3;
            then p1`1 = 0 by A5,A6,Lm4,TOPREAL1:9;
            hence contradiction by A5,A6,A86,EUCLID:57;
           end;
          then {p01} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L1 = {} by A83,TOPREAL1:21,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A82;
        end;
A87:      LSeg(p1,p2) /\ L4 c= {p11} by A58,TOPREAL1:24,XBOOLE_1:26;
          now per cases;
         suppose A88: p2 = p11;
            p2 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L4 <> {} by A88,Lm20,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L4 = {p2} by A87,A88,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A84,ENUMSET1:41;
         suppose A89: p2 <> p11;
             now assume p11 in LSeg(p1,p2) /\ L4;
            then p11 in LSeg(p1,p2) & p1`1 <= p2`1
             by A5,A53,A59,XBOOLE_0:def 3;
            then p11`1 <= p2`1 by TOPREAL1:9;
            then p2`1 = 1 by A53,A54,Lm4,AXIOMS:21;
            hence contradiction by A53,A54,A89,EUCLID:57;
           end;
          then {p11} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L4 = {} by A87,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A84,ENUMSET1:41;
        end;
        hence P1 /\ P2 = {p1,p2};
       suppose A90: q`1 < q1`1;
       take P1 = LSeg(p1,p2),P2 = LSeg(p1,p11) \/ (L4 \/ L3 \/ L1 \/
 LSeg(p01,p2));
       thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
 A91:    now assume
A92:      LSeg(p1,p11) /\ LSeg(p01,p2) <> {};
         consider a being Element of LSeg(p1,p11) /\ LSeg(p01,p2);
         reconsider p = a as Point of TOP-REAL 2 by A92,TARSKI:def 3;
           p in LSeg(p1,p11) & p in LSeg(p01,p2) & p1`1 <= p11`1 & p01`1 <= p2
`1
          by A5,A6,A53,A54,A92,EUCLID:56,XBOOLE_0:def 3;
         then p1`1 <= p`1 & p`1 <= p2`1 by TOPREAL1:9;
         hence contradiction by A5,A53,A90,AXIOMS:22;
        end;
A93:     LSeg(p1,p11) /\ L1 c= L2 /\ L1 by A8,XBOOLE_1:26;
          now assume p01 in LSeg(p1,p11) /\ L1;
         then p01 in LSeg(p1,p11) & p1`1 <=
 p11`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         hence contradiction by A5,A54,A90,Lm4,TOPREAL1:9;
        end;
       then {p01} <> LSeg(p1,p11) /\ L1 by ZFMISC_1:37;
 then A94:   LSeg(p1,p11) /\ L1 = {} by A93,TOPREAL1:21,ZFMISC_1:39;
         LSeg(p1,p11) /\ L3 c= L2 /\ L3 & L3 /\ L2 = {}
        by A8,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A95:   LSeg(p1,p11) /\ L3 = {} by XBOOLE_1:3;
  A96:   LSeg(p1,p11) /\ L4 c= {p11} by A8,TOPREAL1:24,XBOOLE_1:26;
         p11 in LSeg(p1,p11) by TOPREAL1:6;
       then A97: LSeg(p1,p11) /\ L4 <> {} by Lm20,XBOOLE_0:def 3;
 A98:   LSeg(p1,p11) /\ (L4 \/ L3 \/ L1 \/ LSeg(p01,p2))
         = LSeg(p1,p11) /\ (L4 \/ L3 \/ L1) \/ (LSeg(p1,p11) /\ LSeg(p01,p2))
            by XBOOLE_1:23
        .= LSeg(p1,p11) /\ (L4 \/ L3) \/ (LSeg(p1,p11) /\ L1) by A91,XBOOLE_1:
23
        .= LSeg(p1,p11) /\ L4 \/ (LSeg(p1,p11) /\ L3) by A94,XBOOLE_1:23
        .= {p11} by A95,A96,A97,ZFMISC_1:39;
      L4 is_an_arc_of p11,p10 by Lm4,TOPREAL1:15;
 then A99:   L4 \/ L3 is_an_arc_of p11,p00 by TOPREAL1:16,22;
         (L4 \/ L3) /\ L1 = L1 /\ L4 \/ L3 /\ L1 by XBOOLE_1:23
                     .= {p00} by Lm3,TOPREAL1:23;
 then A100:   L4 \/ L3 \/ L1 is_an_arc_of p11,p01 by A99,TOPREAL1:16;
  A101:   L1 /\ LSeg(p01,p2) c= {p01} by A57,TOPREAL1:21,XBOOLE_1:26;
         p01 in LSeg(p01,p2) by TOPREAL1:6;
       then L1 /\ LSeg(p01,p2) <> {} by Lm15,XBOOLE_0:def 3;
 then A102:   L1 /\ LSeg(p01,p2) = {p01} by A101,ZFMISC_1:39;
A103:
       L4 /\ LSeg(p01,p2) c= L4 /\ L2 by A57,XBOOLE_1:26;
          now assume p11 in L4 /\ LSeg(p01,p2);
         then p11 in
 LSeg(p01,p2) & p01`1 <= p2`1 by A53,A54,EUCLID:56,XBOOLE_0:def 3;
         then p11`1 <= p2`1 by TOPREAL1:9;
         hence contradiction by A6,A53,A54,A90,Lm4,AXIOMS:21;
        end;
       then {p11} <> L4 /\ LSeg(p01,p2) by ZFMISC_1:37;
 then A104:   L4 /\ LSeg(p01,p2) = {} by A103,TOPREAL1:24,ZFMISC_1:39;
         L3 /\ LSeg(p01,p2) c= L3 /\ L2 & L3 /\ L2 = {}
        by A57,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A105:   L3 /\ LSeg(p01,p2) = {} by XBOOLE_1:3;
         (L4 \/ L3 \/ L1) /\ LSeg(p01,p2)
         = (L4 \/ L3) /\ LSeg(p01,p2) \/ L1 /\ LSeg(p01,p2) by XBOOLE_1:23
        .= (L4 /\ LSeg(p01,p2)) \/ (L3 /\ LSeg(p01,p2)) \/
 {p01} by A102,XBOOLE_1:23
        .= {p01} by A104,A105;
       then L4 \/ L3 \/ L1 \/
 LSeg(p01,p2) is_an_arc_of p11,p2 by A100,TOPREAL1:16
;
       hence P2 is_an_arc_of p1,p2 by A98,TOPREAL1:17;
       thus P1 \/ P2
         = LSeg(p2,p1) \/ LSeg(p1,p11) \/ (L4 \/ L3 \/ L1 \/
 LSeg(p01,p2)) by XBOOLE_1:4
        .= LSeg(p01,p2) \/ (LSeg(p2,p1) \/ LSeg(p1,p11)) \/ (L4 \/ L3 \/ L1)
            by XBOOLE_1:4
        .= L2 \/ (L4 \/ L3 \/ L1) by A3,A52,TOPREAL1:13
        .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
          p1 in LSeg(p1,p2) & p1 in LSeg(p1,p11) by TOPREAL1:6;
        then p1 in LSeg(p1,p2) /\ LSeg(p1,p11) by XBOOLE_0:def 3;
  then A106:    {p1} c= LSeg(p1,p2) /\ LSeg(p1,p11) by ZFMISC_1:37;
          LSeg(p1,p2) /\ LSeg(p1,p11) c= {p1}
         proof
          let a; assume A107: a in LSeg(p1,p2) /\ LSeg(p1,p11);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p2,p1) & p in LSeg(p1,p11) & p2`2 <= p1`2 & p2`1 <= p1`1
           & p1`1 <= p11`1 by A5,A6,A53,A54,A90,A107,EUCLID:56,XBOOLE_0:def 3;
          then p2`2 <= p`2 & p`2 <= p1`2 & p1`1 <= p`1 & p`1 <= p1`1
           by TOPREAL1:9,10;
          then p1`1 = p`1 & p`2 = 1 by A5,A6,A53,A54,AXIOMS:21;
          then p = |[p1`1, 1]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
          hence a in {p1} by TARSKI:def 1;
         end;
 then A108:    LSeg(p1,p2) /\ LSeg(p1,p11) = {p1} by A106,XBOOLE_0:def 10;
          p2 in LSeg(p1,p2) & p2 in LSeg(p01,p2) by TOPREAL1:6;
        then p2 in LSeg(p1,p2) /\ LSeg(p01,p2) by XBOOLE_0:def 3;
  then A109:    {p2} c= LSeg(p1,p2) /\ LSeg(p01,p2) by ZFMISC_1:37;
          LSeg(p1,p2) /\ LSeg(p01,p2) c= {p2}
         proof
          let a; assume A110: a in LSeg(p1,p2) /\ LSeg(p01,p2);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p2,p1) & p in LSeg(p01,p2) & p2`1 <= p1`1 & p2`2 <= p1`2
           & p01`1 <= p2`1 by A5,A6,A53,A54,A90,A110,EUCLID:56,XBOOLE_0:def 3;
          then p2`2 <= p`2 & p`2 <= p1`2 & p2`1 <= p`1 & p`1 <= p2`1
           by TOPREAL1:9,10;
          then p2`1 = p`1 & p`2 = 1 by A5,A6,A53,A54,AXIOMS:21;
          then p = |[ p2`1, 1]| by EUCLID:57 .= p2 by A53,A54,EUCLID:57;
          hence a in {p2} by TARSKI:def 1;
         end;
 then A111:    LSeg(p1,p2) /\ LSeg(p01,p2) = {p2} by A109,XBOOLE_0:def 10;
          LSeg(p1,p2) /\ L3 c= L2 /\ L3 & L3 /\ L2 = {}
         by A58,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A112:    LSeg(p1,p2) /\ L3 = {} by XBOOLE_1:3;
  A113:    P1 /\ P2
         = {p1} \/ LSeg(p1,p2) /\ (L4 \/ L3 \/ L1 \/
 LSeg(p01,p2)) by A108,XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L3 \/ L1)) \/ {p2}) by A111,XBOOLE_1
:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L3)) \/ (LSeg(p1,p2) /\ L1) \/ {p2})
            by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ (LSeg(p1,p2) /\ L3)
            \/ (LSeg(p1,p2) /\ L1) \/ {p2}) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/ {p2}))
            by A112,XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/
 {p2}) by XBOOLE_1:4;
A114:        LSeg(p1,p2) /\ L1 c= L2 /\ L1 by A58,XBOOLE_1:26;
  A115:    now per cases;
         suppose A116: p2 = p01;
            p2 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L1 <> {} by A116,Lm15,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L1 = {p2} by A114,A116,TOPREAL1:21,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A113;
         suppose A117: p2 <> p01;
             now assume p01 in LSeg(p1,p2) /\ L1;
            then p01 in LSeg(p2,p1) & p2`1 <= p1`1
             by A5,A53,A90,XBOOLE_0:def 3;
            then p2`1 = 0 by A53,A54,Lm4,TOPREAL1:9;
            hence contradiction by A53,A54,A117,EUCLID:57;
           end;
          then {p01} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L1 = {} by A114,TOPREAL1:21,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A113;
        end;
A118:      LSeg(p1,p2) /\ L4 c= {p11} by A58,TOPREAL1:24,XBOOLE_1:26;
          now per cases;
         suppose A119: p1 = p11;
            p1 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L4 <> {} by A119,Lm20,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L4 = {p1} by A118,A119,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A115,ENUMSET1:41;
         suppose A120: p1 <> p11;
             now assume p11 in LSeg(p1,p2) /\ L4;
            then p11 in LSeg(p2,p1) & p2`1 <= p1`1
             by A5,A53,A90,XBOOLE_0:def 3;
            then p11`1 <= p1`1 by TOPREAL1:9;
            then p1`1 = 1 by A5,A6,Lm4,AXIOMS:21;
            hence contradiction by A5,A6,A120,EUCLID:57;
           end;
          then {p11} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L4 = {} by A118,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A115,ENUMSET1:41;
        end;
       hence P1 /\ P2 = {p1,p2};
      end;
     hence ex P1,P2 being non empty Subset of TOP-REAL 2
     st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
      R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2};
    suppose A121: p2 in L3;
     then A122: ex q2 st q2 = p2 & q2`1 <= 1 & q2`1 >= 0 & q2`2 = 0 by TOPREAL1
:19;
     take P1 = LSeg(p1,p11) \/ L4 \/ LSeg(p10,p2),P2
 = LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2);
       LSeg(p10,p2) c= L3 & p10 in LSeg(p10,p2)
 by A121,Lm17,TOPREAL1:6,12;
     then LSeg(p11,p10) /\ LSeg(p10,p2) c= L4 /\ L3 &
      p10 in LSeg(p11,p10) /\ LSeg(p10,p2) by Lm18,XBOOLE_0:def 3,XBOOLE_1:27;
     then LSeg(p11,p10) /\ LSeg(p10,p2) = {p10} by TOPREAL1:22,ZFMISC_1:39;
     then A123: L4 \/ LSeg(p10,p2) is_an_arc_of p11,p2 by Lm4,TOPREAL1:18;
A124: LSeg(p2,p10) c= L3 by A121,Lm17,TOPREAL1:12;
      then LSeg(p1,p11) /\ LSeg(p10,p2) c= L2 /\ L3 by A8,XBOOLE_1:27;
then A125: LSeg(p1,p11) /\ LSeg(p10,p2) = {} by Lm2,XBOOLE_1:3;
        LSeg(p1,p11) /\ (L4 \/ LSeg(p10,p2))
      = (LSeg(p1,p11) /\ L4) \/ (LSeg(p1,p11) /\ LSeg(p10,p2)) by XBOOLE_1:23
     .= {p11} by A13,A14,A125,ZFMISC_1:39;
     then LSeg(p1,p11) \/ (L4 \/ LSeg(p10,p2)) is_an_arc_of p1,p2
      by A123,TOPREAL1:17;
     hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4;
       LSeg(p00,p2) c= L3 & p00 in LSeg(p00,p2)
 by A121,Lm14,TOPREAL1:6,12;
     then LSeg(p01,p00) /\ LSeg(p00,p2) c= {p00} & LSeg(p01,p00) /\
 LSeg(p00,p2) <> {}
      by Lm13,TOPREAL1:23,XBOOLE_0:def 3,XBOOLE_1:27;
     then LSeg(p01,p00) /\ LSeg(p00,p2) = {p00} by ZFMISC_1:39;
     then A126: L1 \/ LSeg(p00,p2) is_an_arc_of p01,p2 by Lm4,TOPREAL1:18;
A127: LSeg(p2,p00) c= L3 by A121,Lm14,TOPREAL1:12;
      then LSeg(p1,p01) /\ LSeg(p00,p2) c= L2 /\ L3 by A7,XBOOLE_1:27;
then A128: LSeg(p1,p01) /\ LSeg(p00,p2) = {} by Lm2,XBOOLE_1:3;
        LSeg(p1,p01) /\ (L1 \/ LSeg(p00,p2))
      = (LSeg(p01,p1) /\ L1) \/ (LSeg(p1,p01) /\ LSeg(p00,p2)) by XBOOLE_1:23
     .= {p01} by A10,A11,A128,TOPREAL1:21,ZFMISC_1:39;
     then LSeg(p1,p01) \/ (L1 \/ LSeg(p00,p2)) is_an_arc_of p1,p2
      by A126,TOPREAL1:17;
     hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4;
 A129: LSeg(p10,p2) \/ LSeg(p00,p2) = L3 by A121,TOPREAL1:11;
 A130: LSeg(p1,p11) \/ LSeg(p1,p01) = L2 by A3,TOPREAL1:11;
     thus R^2-unit_square = L2 \/ (L4 \/ (LSeg(p10,p2) \/ LSeg(p00,p2)) \/
 L1) by A129,TOPREAL1:20,XBOOLE_1:4
      .= L2 \/ (L4 \/ LSeg(p10,p2) \/ LSeg(p00,p2) \/ L1) by XBOOLE_1:4
      .= L2 \/ (L4 \/ LSeg(p10,p2) \/ (L1 \/ LSeg(p00,p2))) by XBOOLE_1:4
      .= LSeg(p1,p11) \/ ((L4 \/ LSeg(p10,p2) \/ (L1 \/ LSeg(p00,p2)))
          \/ LSeg(p1,p01)) by A130,XBOOLE_1:4
      .= LSeg(p1,p11) \/ (L4 \/ LSeg(p10,p2) \/ (L1 \/ LSeg(p00,p2)
          \/ LSeg(p1,p01))) by XBOOLE_1:4
      .= LSeg(p1,p11) \/ (L4 \/ LSeg(p10,p2)) \/ (L1 \/ LSeg(p00,p2)
          \/ LSeg(p1,p01)) by XBOOLE_1:4
      .= LSeg(p1,p11) \/ L4 \/ LSeg(p10,p2)
          \/ (LSeg(p1,p01) \/ (L1 \/ LSeg(p00,p2))) by XBOOLE_1:4
      .= P1 \/ P2 by XBOOLE_1:4;
       LSeg(p1,p11) /\ LSeg(p00,p2) c= L2 /\ LSeg(p00,p2) &
      L2 /\ LSeg(p00,p2) c= L2 /\ L3 by A8,A127,XBOOLE_1:26;
     then LSeg(p1,p11) /\ LSeg(p00,p2) c= L3 /\ L2 by XBOOLE_1:1;
 then A131: LSeg(p1,p11) /\ LSeg(p00,p2) = {} by Lm2,XBOOLE_1:3;
A132: LSeg(p1,p11) /\ LSeg(p1,p01) = {p1} by A3,TOPREAL1:14;
A133: LSeg(p10,p2) /\ LSeg(p00,p2) = {p2} by A121,TOPREAL1:14;
       LSeg(p10,p2) /\ LSeg(p1,p01) c= L3 /\ L2 by A7,A124,XBOOLE_1:27;
 then A134: LSeg(p10,p2) /\ LSeg(p1,p01) = {} by Lm2,XBOOLE_1:3;
 A135:  P1 /\ P2
      = (LSeg(p1,p11) \/ L4) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by XBOOLE_1:
23
     .= (LSeg(p1,p11) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2)))
         \/ (L4 /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by XBOOLE_1:
23
     .= (LSeg(p1,p11) /\ (LSeg(p1,p01) \/ L1))
         \/ (LSeg(p1,p11) /\ LSeg(p00,p2))
         \/ (L4 /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by XBOOLE_1:
23
     .= (LSeg(p1,p11) /\ LSeg(p1,p01)) \/ (LSeg(p1,p11) /\ L1)
         \/ (L4 /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/
 LSeg(p00,p2))) by A131,XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p11) /\ L1)
         \/ ((L4 /\ (LSeg(p1,p01) \/ L1)) \/ (L4 /\ LSeg(p00,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/
 LSeg(p00,p2))) by A132,XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p11) /\ L1)
         \/ ((L4 /\ LSeg(p1,p01)) \/ (L1 /\ L4) \/ (L4 /\ LSeg(p00,p2)))
         \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by XBOOLE_1:
23
     .= {p1} \/ (LSeg(p1,p11) /\ L1)
         \/ ((L4 /\ LSeg(p1,p01)) \/ (L4 /\ LSeg(p00,p2)))
         \/ ((LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1)) \/ {p2}) by A133,Lm3,
XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p11) /\ L1)
         \/ ((L4 /\ LSeg(p1,p01)) \/ (L4 /\ LSeg(p00,p2)))
         \/ ((LSeg(p10,p2) /\ LSeg(p1,p01)) \/ (LSeg(p10,p2) /\ L1) \/ {p2})
          by XBOOLE_1:23
     .= {p1} \/ (LSeg(p1,p11) /\ L1)
         \/ ((L4 /\ LSeg(p1,p01)) \/ (L4 /\ LSeg(p00,p2)))
         \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A134;
A136:    now per cases;
       suppose A137: p1 = p01;
 then L4 /\ LSeg(p1,p01) = L4 /\ {p01} by TOPREAL1:7
                          .= {} by Lm1,Lm8;
        hence P1 /\ P2 =
 {p1} \/ (L4 /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2})
             by A135,A137,TOPREAL1:21;
       suppose A138: p1 = p11;
 then LSeg(p1,p11) /\ L1 = {p11} /\ L1 by TOPREAL1:7
                          .= {} by Lm1,Lm11;
       hence P1 /\ P2
         = {p1} \/ {p1} \/ (L4 /\ LSeg(p00,p2))
           \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A135,A138,TOPREAL1:24,XBOOLE_1:
4
        .= {p1} \/ (L4 /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2});
       suppose A139: p1 <> p11 & p1 <> p01;
A140:        LSeg(p1,p11) /\ L1 c= L2 /\ L1 by A8,XBOOLE_1:26;

          now assume p01 in LSeg(p1,p11) /\ L1;
         then p01 in LSeg(p1,p11) & p1`1 <=
 p11`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         then p1`1 = 0 by A5,A6,Lm4,TOPREAL1:9;
         hence contradiction by A5,A6,A139,EUCLID:57;
        end;
        then {p01} <> LSeg(p1,p11) /\ L1 by ZFMISC_1:37;
 then A141:    LSeg(p1,p11) /\ L1 = {} by A140,TOPREAL1:21,ZFMISC_1:39;
A142:        L4 /\ LSeg(p1,p01) c= L4 /\ L2 by A7,XBOOLE_1:26;

          now assume p11 in L4 /\ LSeg(p1,p01);
         then p11 in LSeg(p01,p1) & p01`1 <=
 p1`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         then 1 <= p1`1 by Lm4,TOPREAL1:9;
         then p1`1 = 1 by A5,A6,AXIOMS:21;
         hence contradiction by A5,A6,A139,EUCLID:57;
        end;
        then {p11} <> L4 /\ LSeg(p1,p01) by ZFMISC_1:37;
 then L4 /\ LSeg(p1,p01) = {} by A142,TOPREAL1:24,ZFMISC_1:39;
        hence P1 /\ P2 =
 {p1} \/ (L4 /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2})
             by A135,A141;
       end;
         now per cases;
        suppose A143: p2 = p00;
 then L4 /\ LSeg(p00,p2) = L4 /\ {p00} by TOPREAL1:7
                          .= {} by Lm1,Lm5;
         hence P1 /\ P2 = {p1,p2} by A136,A143,ENUMSET1:41,TOPREAL1:23;
         suppose A144: p2 = p10;
      then LSeg(p10,p2) /\ L1 = {p10} /\ L1 by TOPREAL1:7
                       .= {} by Lm1,Lm9;
         hence P1 /\ P2 = {p1} \/ ({p2} \/
 {p2}) by A136,A144,TOPREAL1:22,XBOOLE_1:4
          .= {p1,p2} by ENUMSET1:41;
         suppose A145: p2 <> p10 & p2 <> p00;
A146:          L4 /\ LSeg(p00,p2) c= L4 /\ L3 by A127,XBOOLE_1:26;

           now assume p10 in L4 /\ LSeg(p00,p2);
          then p10 in LSeg(p00,p2) & p00`1 <=
 p2`1 by A122,EUCLID:56,XBOOLE_0:def 3;
          then 1 <= p2`1 by Lm4,TOPREAL1:9;
          then p2`1 = 1 by A122,AXIOMS:21;
          hence contradiction by A122,A145,EUCLID:57;
         end;
         then {p10} <> L4 /\ LSeg(p00,p2) by ZFMISC_1:37;
 then A147:     L4 /\ LSeg(p00,p2) = {} by A146,TOPREAL1:22,ZFMISC_1:39;
A148:     LSeg(p10,p2) /\ L1 c= L3 /\ L1 by A124,XBOOLE_1:26;

           now assume p00 in LSeg(p10,p2) /\ L1;
          then p00 in LSeg(p2,p10) & p2`1 <=
 p10`1 by A122,EUCLID:56,XBOOLE_0:def 3;
          then p2`1 = 0 by A122,Lm4,TOPREAL1:9;
          hence contradiction by A122,A145,EUCLID:57;
         end;
         then {p00} <> LSeg(p10,p2) /\ L1 by ZFMISC_1:37;
 then LSeg(p10,p2) /\ L1 = {} by A148,TOPREAL1:23,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A136,A147,ENUMSET1:41;
       end;
     hence P1 /\ P2 = {p1,p2};
    suppose A149: p2 in L4;
     then A150: ex q st q = p2 & q`1 = 1 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:19;
    take P1 = LSeg(p1,p11) \/ LSeg(p11,p2),P2
 = LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2));
      p11 in LSeg(p1,p11) & p11 in LSeg(p11,p2) by TOPREAL1:6;
    then LSeg(p1,p11) c= L2 & LSeg(p11,p2) c= L4 &
     p11 in LSeg(p1,p11) /\ LSeg(p11,p2) by A3,A149,Lm19,Lm20,TOPREAL1:12,
XBOOLE_0:def 3;
    then LSeg(p1,p11) /\ LSeg(p11,p2) c= L2 /\ L4 & L2 /\ L4 = {p11}
     & LSeg(p1,p11) /\ LSeg(p11,p2) <> {} by TOPREAL1:24,XBOOLE_1:27;
    then LSeg(p1,p11) /\ LSeg(p11,p2) = {p11} & (p1 <> p11 or p11 <> p2)
     by A1,ZFMISC_1:39;
    hence P1 is_an_arc_of p1,p2 by TOPREAL1:18;
      L1 is_an_arc_of p01,p00 & L3 is_an_arc_of p00,p10
     by Lm4,TOPREAL1:15;
then A151: L1 \/ L3 is_an_arc_of p01,p10 by TOPREAL1:5,23;
A152: LSeg(p10,p2) c= L4 by A149,Lm18,TOPREAL1:12;
    then L1 /\ LSeg(p10,p2) c= L1 /\ L4 by XBOOLE_1:26;
then A153: L1 /\ LSeg(p10,p2) = {} by Lm3,XBOOLE_1:3;
      p10 in LSeg(p10,p2) by TOPREAL1:6;
    then A154: L3 /\ LSeg(p10,p2) c= {p10} & L3 /\ LSeg(p10,p2) <> {}
     by A152,Lm17,TOPREAL1:22,XBOOLE_0:def 3,XBOOLE_1:27;
      (L1 \/ L3) /\ LSeg(p10,p2)
     = (L1 /\ LSeg(p10,p2)) \/ (L3 /\ LSeg(p10,p2)) by XBOOLE_1:23
    .= {p10} by A153,A154,ZFMISC_1:39;
then A155: L1 \/ L3 \/ LSeg(p10,p2) is_an_arc_of p01,p2 by A151,TOPREAL1:16;
    A156: LSeg(p1,p01) /\
 LSeg(p10,p2) c= {p11} by A7,A152,TOPREAL1:24,XBOOLE_1:27;
      now assume p11 in LSeg(p1,p01) /\ LSeg(p10,p2);
     then p11 in LSeg(p01,p1) & p11 in LSeg(p10,p2) & p01`1 <= p1`1 &
      p10`2 <= p2`2 by A5,A6,A150,EUCLID:56,XBOOLE_0:def 3;
     then p11`1 <= p1`1 & p11`2 <= p2`2 by TOPREAL1:9,10;
     then A157: p11`2 = p1`2 & p11`2 = p2`2 & p11`1 = p1`1 & p11`1 = p2`1
      by A5,A6,A150,Lm4,AXIOMS:21;
     then p1 = |[p11`1,p11`2]| by EUCLID:57 .= p2 by A157,EUCLID:57;
     hence contradiction by A1;
    end;
    then {p11} <> LSeg(p1,p01) /\ LSeg(p10,p2) by ZFMISC_1:37;
then A158: LSeg(p1,p01) /\ LSeg(p10,p2) = {} by A156,ZFMISC_1:39;
      LSeg(p1,p01) /\ (L1 \/ L3 \/ LSeg(p10,p2))
      = (LSeg(p1,p01) /\ (L1 \/ L3)) \/ (LSeg(p1,p01) /\
 LSeg(p10,p2)) by XBOOLE_1:23
     .= (LSeg(p1,p01) /\ L1) \/ (LSeg(p1,p01) /\ L3) by A158,XBOOLE_1:23
     .= {p01} by A9,A10,A11,TOPREAL1:21,ZFMISC_1:39;
    hence P2 is_an_arc_of p1,p2 by A155,TOPREAL1:17;
   A159: L2 = LSeg(p1,p11) \/ LSeg(p1,p01) by A3,TOPREAL1:11;
   A160: L4 = LSeg(p10,p2) \/ LSeg(p11,p2) by A149,TOPREAL1:11;
    thus P1 \/ P2
     = LSeg(p11,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p01) \/
        (L1 \/ L3 \/ LSeg(p10,p2)))) by XBOOLE_1:4
    .= L2 \/ (L1 \/ L3 \/ LSeg(p10,p2)) \/ LSeg(p11,p2) by A159,XBOOLE_1:4
    .= L2 \/ ((L1 \/ L3 \/ LSeg(p10,p2)) \/ LSeg(p11,p2)) by XBOOLE_1:4
    .= L2 \/ (L1 \/ L3 \/ (LSeg(p10,p2) \/ LSeg(p11,p2))) by XBOOLE_1:4
    .= L2 \/ (L1 \/ (L3 \/ L4)) by A160,XBOOLE_1:4
    .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
A161: LSeg(p2,p11) c= L4 by A149,Lm20,TOPREAL1:12;
A162: {p1} = LSeg(p1,p11) /\ LSeg(p1,p01) by A3,TOPREAL1:14;
A163: LSeg(p11,p2) /\ LSeg(p10,p2)
         = {p2} by A149,TOPREAL1:14;
      LSeg(p11,p2) /\ L1 c= L4 /\ L1 by A161,XBOOLE_1:27;
then A164: LSeg(p11,p2) /\ L1 = {} by Lm3,XBOOLE_1:3;
A165:  P1 /\ P2 = (LSeg(p1,p11) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/
 LSeg(p10,p2))))
         \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2))))
         by XBOOLE_1:23
     .= (LSeg(p1,p11) /\ LSeg(p1,p01))
        \/ (LSeg(p1,p11) /\ (L1 \/ L3 \/ LSeg(p10,p2)))
        \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ (L1 \/ L3)) \/ (LSeg(p1,p11) /\
 LSeg(p10,p2)))
        \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2))))
         by A162,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3)
        \/ (LSeg(p1,p11) /\ LSeg(p10,p2)))
        \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3)
        \/ (LSeg(p1,p11) /\ LSeg(p10,p2)))
        \/ ((LSeg(p11,p2) /\ LSeg(p1,p01))
        \/ (LSeg(p11,p2) /\ (L1 \/ L3 \/ LSeg(p10,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3)
        \/ (LSeg(p1,p11) /\ LSeg(p10,p2)))
        \/ ((LSeg(p11,p2) /\ LSeg(p1,p01))
        \/ ((LSeg(p11,p2) /\ (L1 \/ L3)) \/ {p2})) by A163,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3)
        \/ (LSeg(p1,p11) /\ LSeg(p10,p2)))
        \/ ((LSeg(p11,p2) /\ LSeg(p1,p01))
        \/ ((LSeg(p11,p2) /\ L1) \/ (LSeg(p11,p2) /\ L3) \/ {p2})) by XBOOLE_1:
23
     .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ LSeg(p10,p2)))
        \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/ (LSeg(p11,p2) /\ L3 \/ {p2}))
         by A12,A164;
A166:  now per cases;
     suppose A167: p2 = p10;
then A168:     not p2 in LSeg(p1,p11) by A8,Lm4,TOPREAL1:10;
     LSeg(p1,p11) /\ LSeg(p10,p2) = LSeg(p1,p11) /\ {p2} by A167,TOPREAL1:7
             .= {} by A168,Lm1;
      hence P1 /\ P2
        = {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01))
           \/ {p2}) by A165,A167,TOPREAL1:22;
     suppose A169: p2 = p11;
      then A170: LSeg(p11,p2) /\ L3 = {p11} /\ L3 by TOPREAL1:7
                           .= {} by Lm1,Lm12;
       A171: LSeg(p1,p11) /\ LSeg(p10,p2) c= {p2} by A8,A169,TOPREAL1:24,
XBOOLE_1:27;
            p2 in LSeg(p1,p11) by A169,TOPREAL1:6;
          then LSeg(p1,p11) /\
 LSeg(p10,p2) <> {} by A169,Lm20,XBOOLE_0:def 3;
      then LSeg(p1,p11) /\ LSeg(p10,p2) = {p2} by A171,ZFMISC_1:39;
      hence P1 /\ P2
         = ({p1} \/ (LSeg(p1,p11) /\ L1)) \/ {p2} \/ ((LSeg(p11,p2) /\
 LSeg(p1,p01))
             \/ {p2}) by A165,A170,XBOOLE_1:4
        .= ({p1} \/ (LSeg(p1,p11) /\ L1)) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01))
             \/ {p2} \/ {p2}) by XBOOLE_1:4
        .= ({p1} \/ (LSeg(p1,p11) /\ L1)) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01))
             \/ ({p2} \/ {p2})) by XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/
 {p2});
     suppose A172: p2 <> p11 & p2 <> p10;
A173:
      LSeg(p11,p2) /\ L3 c= L4 /\ L3 by A161,XBOOLE_1:27;
         now assume p10 in LSeg(p11,p2) /\ L3;
        then p10 in LSeg(p2,p11) & p2`2 <= p11`2 by A150,EUCLID:56,XBOOLE_0:def
3;
        then p2`1 = 1 & p2`2 = 0 by A150,Lm4,TOPREAL1:10;
        hence contradiction by A172,EUCLID:57;
       end;
       then {p10} <> LSeg(p11,p2) /\ L3 by ZFMISC_1:37;
     then A174: LSeg(p11,p2) /\ L3 = {} by A173,TOPREAL1:22,ZFMISC_1:39;
      A175: LSeg(p1,p11) /\ LSeg(p10,p2) c= {p11} by A8,A152,TOPREAL1:24,
XBOOLE_1:27;
          now assume p11 in LSeg(p1,p11) /\ LSeg(p10,p2);
         then p11 in
 LSeg(p10,p2) & p10`2 <= p2`2 by A150,EUCLID:56,XBOOLE_0:def 3;
         then p11`2 <= p2`2 by TOPREAL1:10;
         then p2`1 = 1 & p2`2 = 1 by A150,Lm4,AXIOMS:21;
         hence contradiction by A172,EUCLID:57;
        end;
       then {p11} <> LSeg(p1,p11) /\ LSeg(p10,p2) by ZFMISC_1:37;
     then LSeg(p1,p11) /\ LSeg(p10,p2) = {} by A175,ZFMISC_1:39;
     hence P1 /\ P2
         = {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/
 {p2})
            by A165,A174;
      end;
        now per cases;
       suppose A176: p1 = p01;
        then A177:  LSeg(p11,p2) /\ LSeg(p1,p01) = LSeg(p11,p2) /\ {p1} by
TOPREAL1:7;
                p1 in LSeg(p11,p2) implies
               contradiction by A161,A176,Lm4,TOPREAL1:9;
          then LSeg(p11,p2) /\ LSeg(p1,p01) = {} by A177,Lm1;
        hence P1 /\ P2
           = {p1,p2} by A166,A176,ENUMSET1:41,TOPREAL1:21;
       suppose A178: p1 = p11;
           then LSeg(p1,p11) /\ L1 = {p1} /\ L1 by TOPREAL1:7;
             then A179: LSeg(p1,p11) /\ L1 = {} by A178,Lm1,Lm11;
A180:             LSeg(p11,p2) /\ LSeg(p1,p01) c= L4 /\ L2 by A7,A161,XBOOLE_1:
27;
               p11 in LSeg(p11,p2) by TOPREAL1:6;
             then LSeg(p11,p2) /\ LSeg(p1,p01) <> {} by A178,Lm19,XBOOLE_0:def
3
;
         then LSeg(p11,p2) /\ LSeg(p1,p01) = {p1} by A178,A180,TOPREAL1:24,
ZFMISC_1:39;
        hence P1 /\ P2
          = {p1} \/ {p1} \/ {p2} by A166,A179,XBOOLE_1:4
         .= {p1,p2} by ENUMSET1:41;
       suppose A181: p1 <> p11 & p1 <> p01;
A182:            LSeg(p1,p11) /\ L1 c= L2 /\ L1 by A8,XBOOLE_1:27;
               now assume p01 in LSeg(p1,p11) /\ L1;
              then p01 in LSeg(p1,p11) & p1`1 <= p11`1
               by A5,A6,EUCLID:56,XBOOLE_0:def 3;
              then p1`1 = 0 by A5,A6,Lm4,TOPREAL1:9;
              hence contradiction by A5,A6,A181,EUCLID:57;
             end;
             then {p01} <> LSeg(p1,p11) /\ L1 by ZFMISC_1:37;
         then A183: LSeg(p1,p11) /\ L1 = {} by A182,TOPREAL1:21,ZFMISC_1:39;
A184:             LSeg(p11,p2) /\ LSeg(p1,p01) c= L4 /\ L2
              by A7,A161,XBOOLE_1:27;
               now assume p11 in LSeg(p11,p2) /\ LSeg(p1,p01);
              then p11 in LSeg(p01,p1) & p01`1 <= p1`1
               by A5,A6,EUCLID:56,XBOOLE_0:def 3;
              then p11`1 <= p1`1 by TOPREAL1:9;
              then p1`1 = 1 by A5,A6,Lm4,AXIOMS:21;
              hence contradiction by A5,A6,A181,EUCLID:57;
             end;
             then {p11} <> LSeg(p11,p2) /\ LSeg(p1,p01) by ZFMISC_1:37;
         then LSeg(p11,p2) /\ LSeg(p1,p01) = {}
 by A184,TOPREAL1:24,ZFMISC_1:39;
        hence P1 /\ P2
          = {p1,p2} by A166,A183,ENUMSET1:41;
      end;
    hence P1 /\ P2 = {p1,p2};
 end;

Lm25:
 p1 <> p2 & p2 in R^2-unit_square &
 p1 in LSeg(p00, p10) implies
  ex P1,P2 being non empty Subset of TOP-REAL 2
  st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
   R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2}
 proof assume that
 A1: p1 <> p2 and
 A2: p2 in R^2-unit_square and
 A3: p1 in LSeg(p00, p10);
A4:   p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:20,XBOOLE_0:def 2;

   consider p such that
  A5: p = p1 and A6: p`1 <= 1 & p`1 >= 0 & p`2 = 0 by A3,TOPREAL1:19;
  A7: LSeg(p00,p1) c= L3 by A3,Lm14,TOPREAL1:12;
  A8: LSeg(p10,p1) c= L3 by A3,Lm17,TOPREAL1:12;
      then LSeg(p10,p1) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {}
        by TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
  then A9: LSeg(p10,p1) /\ L2 = {} by XBOOLE_1:3;
        p10 in LSeg(p10,p1) by TOPREAL1:6;
      then A10: LSeg(p10,p1) /\ L4 c= L3 /\ L4 & LSeg(p10,p1) /\ L4 <> {}
       by A8,Lm18,XBOOLE_0:def 3,XBOOLE_1:26;
        p00 in LSeg(p1,p00) by TOPREAL1:6;
      then A11: LSeg(p1,p00) /\ L1 c= L3 /\ L1 &
       p00 in LSeg(p1,p00) /\ L1 by A7,Lm13,XBOOLE_0:def 3,XBOOLE_1:26;
  A12: LSeg(p1,p00) /\ LSeg(p1,p10)
        = {p1} by A3,TOPREAL1:14;
        LSeg(p1,p00) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {}
       by A7,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
  then A13: LSeg(p1,p00) /\ L2 = {} by XBOOLE_1:3;
   per cases by A4,XBOOLE_0:def 2;
    suppose A14: p2 in L1;
     then A15: ex q st q = p2 & q`1 = 0 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:19;
  A16: LSeg(p00,p2) c= L1 by A14,Lm13,TOPREAL1:12;
  A17: LSeg(p2,p01) c= L1 by A14,Lm15,TOPREAL1:12;
     take P1 = LSeg(p1,p00) \/ LSeg(p00,p2),P2
 = LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2));
        p00 in LSeg(p1,p00) & p00 in LSeg(p00,p2) by TOPREAL1:6;
      then LSeg(p1,p00) /\ LSeg(p00,p2) c= L3 /\ L1 &
      p00 in LSeg(p1,p00) /\ LSeg(p00,p2) & L1 /\ L3 = {p00}
       by A7,A16,TOPREAL1:23,XBOOLE_0:def 3,XBOOLE_1:27;
      then LSeg(p1,p00) /\ LSeg(p00,p2) = {p00} & (p1 <> p00 or p00 <> p2)
       by A1,ZFMISC_1:39;
    hence P1 is_an_arc_of p1,p2 by TOPREAL1:18;
        L4 is_an_arc_of p10,p11 & L2 is_an_arc_of p11,p01
      by Lm4,TOPREAL1:15;
     then A18: L4 \/ L2 is_an_arc_of p10,p01 by TOPREAL1:5,24;
       L4 /\ LSeg(p01,p2) c= L4 /\ L1 by A17,XBOOLE_1:26;
 then A19: L4 /\ LSeg(p01,p2) = {} by Lm3,XBOOLE_1:3;
       p01 in LSeg(p01,p2) by TOPREAL1:6;
     then A20: L2 /\ LSeg(p01,p2) c= L2 /\ L1 & p01 in L2 /\ LSeg(p01,p2)
      by A17,Lm16,XBOOLE_0:def 3,XBOOLE_1:27;
       (L4 \/ L2) /\ LSeg(p01,p2)
      = (L4 /\ LSeg(p01,p2)) \/ (L2 /\ LSeg(p01,p2)) by XBOOLE_1:23
     .= {p01} by A19,A20,TOPREAL1:21,ZFMISC_1:39;
 then A21: L4 \/ L2 \/ LSeg(p01,p2) is_an_arc_of p10,p2 by A18,TOPREAL1:16;
A22:    LSeg(p1,p10) /\ LSeg(p01,p2) c= L3 /\ L1
     by A8,A17,XBOOLE_1:27;
       now assume p00 in LSeg(p1,p10) /\ LSeg(p01,p2);
      then p00 in LSeg(p1,p10) & p00 in LSeg(p2,p01) & p1`1 <= p10`1 &
       p2`2 <= p01`2 by A5,A6,A15,EUCLID:56,XBOOLE_0:def 3;
     then A23: 0 = p1`2 & 0 = p2`2 & 0 = p1`1 & 0 = p2`1 by A5,A6,A15,Lm4,
TOPREAL1:9,10;
      then p1 = p00 by EUCLID:57 .= p2 by A23,EUCLID:57;
      hence contradiction by A1;
     end;
     then {p00} <> LSeg(p1,p10) /\ LSeg(p01,p2) by ZFMISC_1:37;
 then A24: LSeg(p1,p10) /\ LSeg(p01,p2) = {} by A22,TOPREAL1:23,ZFMISC_1:39;
       LSeg(p1,p10) /\ (L4 \/ L2 \/ LSeg(p01,p2))
       = (LSeg(p1,p10) /\ (L4 \/ L2)) \/ (LSeg(p1,p10) /\
 LSeg(p01,p2)) by XBOOLE_1:23
      .= (LSeg(p1,p10) /\ L4) \/ (LSeg(p10,p1) /\ L2) by A24,XBOOLE_1:23
      .= {p10} by A9,A10,TOPREAL1:22,ZFMISC_1:39;
     hence P2 is_an_arc_of p1,p2 by A21,TOPREAL1:17;
A25:  LSeg(p1,p00) \/ LSeg(p1,p10) = L3 by A3,TOPREAL1:11;
A26:  LSeg(p01,p2) \/ LSeg(p00,p2) = L1 by A14,TOPREAL1:11;
     thus P1 \/ P2
      = LSeg(p00,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p10) \/
         (L4 \/ L2 \/ LSeg(p01,p2)))) by XBOOLE_1:4
     .= LSeg(p00,p2) \/ (L3 \/ (L4 \/ L2 \/ LSeg(p01,p2))) by A25,XBOOLE_1:4
     .= LSeg(p00,p2) \/ (L3 \/ (L4 \/ L2) \/ LSeg(p01,p2)) by XBOOLE_1:4
     .= LSeg(p00,p2) \/ (L3 \/ L4 \/ L2 \/ LSeg(p01,p2)) by XBOOLE_1:4
     .= LSeg(p00,p2) \/ (L3 \/ L4 \/ (L2 \/ LSeg(p01,p2))) by XBOOLE_1:4
     .= (L2 \/ LSeg(p01,p2) \/ LSeg(p00,p2)) \/ (L3 \/ L4) by XBOOLE_1:4
     .= R^2-unit_square by A26,TOPREAL1:20,XBOOLE_1:4;
A27: {p1} = LSeg(p1,p00) /\ LSeg(p1,p10) by A3,TOPREAL1:14;
      LSeg(p1,p00) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {}
 by A7,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
then A28: LSeg(p1,p00) /\ L2 = {} by XBOOLE_1:3;
A29: {p2} = LSeg(p00,p2) /\ LSeg(p01,p2) by A14,TOPREAL1:14;
      LSeg(p00,p2) /\ L4 c= L1 /\ L4 & L1 /\ L4 = {} by A16,TOPREAL1:26,
XBOOLE_0:def 7,XBOOLE_1:26
;
then A30: LSeg(p00,p2) /\ L4 = {} by XBOOLE_1:3;
A31:  P1 /\ P2 = (LSeg(p1,p00) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2))))
         \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2))))
         by XBOOLE_1:23
     .= (LSeg(p1,p00) /\ LSeg(p1,p10))
        \/ (LSeg(p1,p00) /\ (L4 \/ L2 \/ LSeg(p01,p2)))
        \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ (L4 \/ L2)) \/ (LSeg(p1,p00) /\
 LSeg(p01,p2)))
        \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2))))
         by A27,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ L2)
        \/ (LSeg(p1,p00) /\ LSeg(p01,p2)))
        \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2)))
        \/ ((LSeg(p00,p2) /\ LSeg(p1,p10))
        \/ (LSeg(p00,p2) /\ (L4 \/ L2 \/ LSeg(p01,p2)))) by A28,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2)))
        \/ ((LSeg(p00,p2) /\ (LSeg(p1,p10))
        \/ ((LSeg(p00,p2) /\ (L4 \/ L2)) \/ {p2}))) by A29,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2)))
        \/ ((LSeg(p00,p2) /\ (LSeg(p1,p10))
        \/ (((LSeg(p00,p2) /\ L4) \/ (LSeg(p00,p2) /\ L2)) \/
 {p2}))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2)))
        \/ ((LSeg(p00,p2) /\ LSeg(p1,p10))
        \/ ((LSeg(p00,p2) /\ L2) \/ {p2})) by A30;
A32:   now per cases;
      suppose A33: p1 = p00;
     then LSeg(p1,p00) /\ L4 = {p00} /\ L4 by TOPREAL1:7;
       then A34:    LSeg(p1,p00) /\ L4 = {} by Lm1,Lm5;
          p1 in LSeg(p00,p2) & p1 in LSeg(p1,p10) by A33,TOPREAL1:6;
       then LSeg(p00,p2) /\ LSeg(p1,p10) c= {p1} & LSeg(p00,p2) /\
 LSeg(p1,p10) <> {}
        by A16,A33,TOPREAL1:23,XBOOLE_0:def 3,XBOOLE_1:26;
    then LSeg(p00,p2) /\ LSeg(p1,p10) = {p1} by ZFMISC_1:39;
       hence P1 /\ P2
        = {p1} \/ ({p1} \/ LSeg(p1,p00) /\ LSeg(p01,p2))
         \/ ((LSeg(p00,p2) /\ L2) \/ {p2}) by A31,A34,XBOOLE_1:4
       .= {p1} \/ {p1} \/ LSeg(p1,p00) /\ LSeg(p01,p2)
         \/ ((LSeg(p00,p2) /\ L2) \/ {p2}) by XBOOLE_1:4
       .= {p1} \/ LSeg(p1,p00) /\ LSeg(p01,p2)
         \/ ((LSeg(p00,p2) /\ L2) \/ {p2});
      suppose A35: p1 = p10;
 then A36:    LSeg(p00,p2) /\ LSeg(p1,p10) = LSeg(p00,p2) /\
 {p10} by TOPREAL1:7;
      not p10 in LSeg(p00,p2) by A16,Lm4,TOPREAL1:9;
    then LSeg(p00,p2) /\ LSeg(p1,p10) = {} by A36,Lm1;
       hence P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p00) /\ LSeg(p01,p2))
         \/ ((LSeg(p00,p2) /\ L2) \/ {p2}) by A31,A35,TOPREAL1:22,XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p00) /\ LSeg(p01,p2))
         \/ ((LSeg(p00,p2) /\ L2) \/ {p2});
      suppose A37: p1 <> p10 & p1 <> p00;
 A38:    LSeg(p1,p00) /\ L4 c= {p10} by A7,TOPREAL1:22,XBOOLE_1:26;
          now assume p10 in LSeg(p1,p00) /\ L4;
         then p10 in LSeg(p00,p1) & p00`1 <=
 p1`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         then p10`1 <= p1`1 by TOPREAL1:9;
         then p1`1 = 1 by A5,A6,Lm4,AXIOMS:21;
         hence contradiction by A5,A6,A37,EUCLID:57;
        end;
       then {p10} <> LSeg(p1,p00) /\ L4 by ZFMISC_1:37;
then A39:    LSeg(p1,p00) /\ L4 = {} by A38,ZFMISC_1:39;
 A40:    LSeg(p00,p2) /\ LSeg(p1,p10) c= {p00} by A8,A16,TOPREAL1:23,XBOOLE_1:
27;
         now assume p00 in LSeg(p00,p2) /\ LSeg(p1,p10);
        then p00 in
 LSeg(p1,p10) & p1`1 <= p10`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
        then 0 = p1`1 by A5,A6,Lm4,TOPREAL1:9;
        hence contradiction by A5,A6,A37,EUCLID:57;
       end;
       then {p00} <> LSeg(p00,p2) /\ LSeg(p1,p10) by ZFMISC_1:37;
then LSeg(p00,p2) /\ LSeg(p1,p10) = {} by A40,ZFMISC_1:39;
       hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ LSeg(p01,p2))
           \/ ((LSeg(p00,p2) /\ L2) \/ {p2}) by A31,A39;
      end;
         now per cases;
        suppose A41: p2 = p00;
A42:        LSeg(p1,p00) /\ LSeg(p01,p2) c= L3 /\ L1 by A7,A17,XBOOLE_1:27;
          p00 in LSeg(p1,p00) by TOPREAL1:6;
        then LSeg(p1,p00) /\ LSeg(p01,p2) <> {} by A41,Lm13,XBOOLE_0:def 3
;
then A43:     LSeg(p1,p00) /\
 LSeg(p01,p2) = {p2} by A41,A42,TOPREAL1:23,ZFMISC_1:39;
        LSeg(p00,p2) /\ L2 = {p00} /\ L2 by A41,TOPREAL1:7;
     then LSeg(p00,p2) /\ L2 = {} by Lm1,Lm6;
        hence P1 /\ P2
          = {p1} \/ ({p2} \/ {p2}) by A32,A43,XBOOLE_1:4
         .= {p1,p2} by ENUMSET1:41;
        hence P1 /\ P2 = {p1,p2};
        suppose A44: p2 = p01;
 then A45:     LSeg(p1,p00) /\ LSeg(p01,p2) = LSeg(p1,p00) /\ {p01} by TOPREAL1
:7;
          not p01 in LSeg(p1,p00) by A7,Lm4,TOPREAL1:10;
        then A46: LSeg(p1,p00) /\ LSeg(p01,p2) = {} by A45,Lm1;
        hence P1 /\ P2 = {p1,p2} by A32,A44,ENUMSET1:41,TOPREAL1:21;
        thus P1 /\ P2 = {p1,p2} by A32,A44,A46,ENUMSET1:41,TOPREAL1:21;
        suppose A47: p2 <> p01 & p2 <> p00;
A48:        LSeg(p1,p00) /\ LSeg(p01,p2) c= L3 /\ L1 by A7,A17,XBOOLE_1:27;
           now assume p00 in LSeg(p1,p00) /\ LSeg(p01,p2);
          then p00 in
 LSeg(p2,p01) & p2`2 <= p01`2 by A15,EUCLID:56,XBOOLE_0:def 3;
          then p2`2 = 0 by A15,Lm4,TOPREAL1:10;
          hence contradiction by A15,A47,EUCLID:57;
         end;
        then {p00} <> LSeg(p1,p00) /\ LSeg(p01,p2) by ZFMISC_1:37;
then A49:     LSeg(p1,p00) /\ LSeg(p01,p2) = {} by A48,TOPREAL1:23,ZFMISC_1:39;
 A50:     LSeg(p00,p2) /\ L2 c= {p01} by A16,TOPREAL1:21,XBOOLE_1:26;
           now assume p01 in LSeg(p00,p2) /\ L2;
          then p01 in
 LSeg(p00,p2) & p00`2 <= p2`2 by A15,EUCLID:56,XBOOLE_0:def 3;
          then p01`2 <= p2`2 by TOPREAL1:10;
          then 1 = p2`2 by A15,Lm4,AXIOMS:21;
          hence contradiction by A15,A47,EUCLID:57;
         end;
        then {p01} <> LSeg(p00,p2) /\ L2 by ZFMISC_1:37;
     then A51: LSeg(p00,p2) /\ L2 = {} by A50,ZFMISC_1:39;
        hence P1 /\ P2 = {p1,p2} by A32,A49,ENUMSET1:41;
        thus P1 /\ P2 = {p1,p2} by A32,A49,A51,ENUMSET1:41;
       end;
     hence P1 /\ P2 = {p1,p2};
    suppose A52: p2 in L2;
    then A53: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 1 by TOPREAL1:19;
    take P1 = LSeg(p1,p00) \/ L1 \/ LSeg(p01,p2),P2
 = LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2);
 A54: LSeg(p2,p01) c= L2 by A52,Lm16,TOPREAL1:12;
 A55: LSeg(p2,p11) c= L2 by A52,Lm19,TOPREAL1:12;
       p01 in LSeg(p01,p2) by TOPREAL1:6;
     then L1 /\ LSeg(p01,p2) c= {p01} & L1 /\ LSeg(p01,p2) <> {}
      by A54,Lm15,TOPREAL1:21,XBOOLE_0:def 3,XBOOLE_1:26;
     then L1 /\ LSeg(p01,p2) = {p01} by ZFMISC_1:39;
  then A56: L1 \/ LSeg(p01,p2) is_an_arc_of p00,p2 by Lm4,TOPREAL1:18;
       LSeg(p1,p00) /\ LSeg(p01,p2) c= L3 /\ L2 by A7,A54,XBOOLE_1:27;
  then A57: LSeg(p1,p00) /\ LSeg(p01,p2) = {} by Lm2,XBOOLE_1:3;
       LSeg(p1,p00) /\ (L1 \/ LSeg(p01,p2))
       = (LSeg(p1,p00) /\ L1) \/ (LSeg(p1,p00) /\ LSeg(p01,p2)) by XBOOLE_1:23
      .= {p00} by A11,A57,TOPREAL1:23,ZFMISC_1:39;
     then LSeg(p1,p00) \/ (L1 \/ LSeg(p01,p2)) is_an_arc_of p1,p2
      by A56,TOPREAL1:17;
    hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4;
       L4 /\ LSeg(p11,p2) c= L4 /\ L2 &
      p11 in LSeg(p11,p2) by A55,TOPREAL1:6,XBOOLE_1:26;
     then L4 /\ LSeg(p11,p2) c= {p11} & L4 /\ LSeg(p11,p2) <> {}
      by Lm20,TOPREAL1:24,XBOOLE_0:def 3;
     then L4 /\ LSeg(p11,p2) = {p11} by ZFMISC_1:39;
 then A58: L4 \/ LSeg(p11,p2) is_an_arc_of p10,p2 by Lm4,TOPREAL1:18;
       LSeg(p1,p10) /\ LSeg(p11,p2) c= L3 /\ L2 by A8,A55,XBOOLE_1:27;
 then A59: LSeg(p1,p10) /\ LSeg(p11,p2) = {} by Lm2,XBOOLE_1:3;
       LSeg(p1,p10) /\ (L4 \/ LSeg(p11,p2))
       = (LSeg(p1,p10) /\ L4) \/ (LSeg(p1,p10) /\ LSeg(p11,p2)) by XBOOLE_1:23
      .= {p10} by A10,A59,TOPREAL1:22,ZFMISC_1:39;
     then LSeg(p1,p10) \/ (L4 \/ LSeg(p11,p2)) is_an_arc_of p1,p2
      by A58,TOPREAL1:17;
    hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4;
    thus R^2-unit_square = L1 \/ (LSeg(p01,p2) \/ LSeg(p11,p2)) \/ (L3 \/
 L4) by A52,TOPREAL1:11,20
      .= L1 \/ LSeg(p01,p2) \/ LSeg(p11,p2) \/ (L3 \/ L4) by XBOOLE_1:4
      .= L1 \/ LSeg(p01,p2) \/ ((L3 \/ L4) \/ LSeg(p11,p2)) by XBOOLE_1:4
      .= L1 \/ LSeg(p01,p2) \/ (L3 \/ (L4 \/ LSeg(p11,p2))) by XBOOLE_1:4
      .= L1 \/ LSeg(p01,p2) \/ (LSeg(p1,p00) \/ LSeg(p1,p10)
          \/ (L4 \/ LSeg(p11,p2))) by A3,TOPREAL1:11
      .= L1 \/ LSeg(p01,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p10)
          \/ (L4 \/ LSeg(p11,p2)))) by XBOOLE_1:4
      .= L1 \/ LSeg(p01,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p10)
          \/ L4 \/ LSeg(p11,p2))) by XBOOLE_1:4
      .= (LSeg(p1,p00) \/ (L1 \/ LSeg(p01,p2))) \/ (LSeg(p1,p10)
          \/ L4 \/ LSeg(p11,p2)) by XBOOLE_1:4
      .= P1 \/ P2 by XBOOLE_1:4;
A60:  LSeg(p01,p2) /\ LSeg(p11,p2)
       = {p2} by A52,TOPREAL1:14;
       LSeg(p01,p2) /\ LSeg(p1,p10) c= L2 /\ L3 by A8,A54,XBOOLE_1:27;
then A61:  LSeg(p01,p2) /\ LSeg(p1,p10) = {} by Lm2,XBOOLE_1:3;
       LSeg(p1,p00) /\ LSeg(p11,p2) c= L3 /\ L2 & L3 /\ L2 = {}
      by A7,A55,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:27;
then A62:  LSeg(p1,p00) /\ LSeg(p11,p2) = {} by XBOOLE_1:3;
A63:   P1 /\ P2
       = (LSeg(p1,p00) \/ L1) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))) by XBOOLE_1:
23
      .= (LSeg(p1,p00) \/ L1) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))
        \/ (LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L4) \/ {p2}) by A60,XBOOLE_1:23
      .= (LSeg(p1,p00) \/ L1) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))
        \/ ((LSeg(p01,p2) /\ LSeg(p1,p10)) \/ (LSeg(p01,p2) /\ L4) \/ {p2})
         by XBOOLE_1:23
      .= (LSeg(p1,p00) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2)))
        \/ (L1 /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A61,XBOOLE_1:23
      .= ((LSeg(p1,p00) /\ (LSeg(p1,p10) \/ L4))
        \/ (LSeg(p1,p00) /\ LSeg(p11,p2)))
        \/ (L1 /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by XBOOLE_1:23
      .= ({p1} \/ (LSeg(p1,p00) /\ L4))
        \/ (L1 /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A12,A62,XBOOLE_1:23
      .= ({p1} \/ (LSeg(p1,p00) /\ L4))
        \/ ((L1 /\ (LSeg(p1,p10) \/ L4)) \/ (L1 /\ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by XBOOLE_1:23
      .= ({p1} \/ (LSeg(p1,p00) /\ L4))
        \/ ((L1 /\ LSeg(p1,p10)) \/ (L1 /\ L4) \/ (L1 /\ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by XBOOLE_1:23
      .= ({p1} \/ (LSeg(p1,p00) /\ L4))
        \/ ((L1 /\ LSeg(p1,p10)) \/ (L1 /\ LSeg(p11,p2)))
        \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by Lm3;
A64:   now per cases;
      suppose A65: p1 = p00;
    then LSeg(p1,p00) /\ L4 = {p00} /\ L4 by TOPREAL1:7;
   then LSeg(p1,p00) /\ L4 = {} by Lm1,Lm5;
      hence P1 /\ P2 = {p1} \/ {p1} \/ (L1 /\ LSeg(p11,p2))
          \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A63,A65,TOPREAL1:23,XBOOLE_1:4
        .= {p1} \/ (L1 /\ LSeg(p11,p2)) \/ ((LSeg(p01,p2) /\ L4) \/ {p2});
      suppose A66: p1 = p10;
    then L1 /\ LSeg(p1,p10) = L1 /\ {p10} by TOPREAL1:7;
   then L1 /\ LSeg(p1,p10) = {} by Lm1,Lm9;
      hence P1 /\ P2 = {p1} \/ (L1 /\ LSeg(p11,p2)) \/ ((LSeg(p01,p2) /\ L4)
\/
 {p2})
 by A63,A66,TOPREAL1:22;
      suppose A67: p1 <> p10 & p1 <> p00;
 A68:   LSeg(p1,p00) /\ L4 c= {p10} by A7,TOPREAL1:22,XBOOLE_1:26;
         now assume p10 in LSeg(p1,p00) /\ L4;
        then p10 in
 LSeg(p00,p1) & p00`1 <= p1`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
        then p10`1 <= p1`1 by TOPREAL1:9;
        then 1 = p1`1 by A5,A6,Lm4,AXIOMS:21;
        hence contradiction by A5,A6,A67,EUCLID:57;
       end;
      then {p10} <> LSeg(p1,p00) /\ L4 by ZFMISC_1:37;
then A69:   LSeg(p1,p00) /\ L4 = {} by A68,ZFMISC_1:39;
 A70:   L1 /\ LSeg(p1,p10) c= {p00} by A8,TOPREAL1:23,XBOOLE_1:26;
         now assume p00 in L1 /\ LSeg(p1,p10);
        then p00 in
 LSeg(p1,p10) & p1`1 <= p10`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
        then 0 = p1`1 by A5,A6,Lm4,TOPREAL1:9;
        hence contradiction by A5,A6,A67,EUCLID:57;
       end;
      then {p00} <> L1 /\ LSeg(p1,p10) by ZFMISC_1:37;
   then L1 /\ LSeg(p1,p10) = {} by A70,ZFMISC_1:39;
      hence P1 /\ P2 = {p1} \/ (L1 /\ LSeg(p11,p2)) \/ ((LSeg(p01,p2) /\ L4)
\/
 {p2})
 by A63,A69;
      end;
         now per cases;
        suppose A71: p2 = p01;
      then LSeg(p01,p2) /\ L4 = {p01} /\ L4 by TOPREAL1:7;
        then LSeg(p01,p2) /\ L4 = {} by Lm1,Lm8;
        hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A64,A71,TOPREAL1:21,XBOOLE_1
:4
          .= {p1,p2} by ENUMSET1:41;
        suppose A72: p2 = p11;
      then L1 /\ LSeg(p11,p2) = L1 /\ {p11} by TOPREAL1:7;
     then L1 /\ LSeg(p11,p2) = {} by Lm1,Lm11;
        hence P1 /\ P2 = {p1,p2} by A64,A72,ENUMSET1:41,TOPREAL1:24;
        suppose A73: p2 <> p11 & p2 <> p01;
 A74:     L1 /\ LSeg(p11,p2) c= {p01} by A55,TOPREAL1:21,XBOOLE_1:26;
           now assume p01 in L1 /\ LSeg(p11,p2);
          then p01 in
 LSeg(p2,p11) & p2`1 <= p11`1 by A53,EUCLID:56,XBOOLE_0:def 3;
          then p2`1 = 0 by A53,Lm4,TOPREAL1:9;
          hence contradiction by A53,A73,EUCLID:57;
         end;
        then {p01} <> L1 /\ LSeg(p11,p2) by ZFMISC_1:37;
then A75:     L1 /\ LSeg(p11,p2) = {} by A74,ZFMISC_1:39;
 A76:     LSeg(p01,p2) /\ L4 c= {p11} by A54,TOPREAL1:24,XBOOLE_1:26;
           now assume p11 in LSeg(p01,p2) /\ L4;
          then p11 in
 LSeg(p01,p2) & p01`1 <= p2`1 by A53,EUCLID:56,XBOOLE_0:def 3;
          then p11`1 <= p2`1 by TOPREAL1:9;
          then 1 = p2`1 by A53,Lm4,AXIOMS:21;
          hence contradiction by A53,A73,EUCLID:57;
         end;
        then {p11} <> LSeg(p01,p2) /\ L4 by ZFMISC_1:37;
     then LSeg(p01,p2) /\ L4 = {} by A76,ZFMISC_1:39;
        hence P1 /\ P2 = {p1,p2} by A64,A75,ENUMSET1:41;
       end;
    hence P1 /\ P2 = {p1,p2};
    suppose A77: p2 in L3;
     then consider q such that
  A78: q = p2 and A79: q`1 <= 1 & q`1 >= 0 & q`2 = 0 by TOPREAL1:19;
A80:     p = |[p`1,p`2]| & q = |[q`1,q`2]| by EUCLID:57;
 A81: LSeg(p2,p10) c= L3 by A77,Lm17,TOPREAL1:12;
 A82: LSeg(p2,p00) c= L3 by A77,Lm14,TOPREAL1:12;
 A83: LSeg(p1,p2) c= L3 by A3,A77,TOPREAL1:12;
        now per cases by A1,A5,A6,A78,A79,A80,REAL_1:def 5;
       suppose A84: p`1 < q`1;
       take P1 = LSeg(p1,p2),P2 = LSeg(p1,p00) \/ (L1 \/ L2 \/ L4 \/
 LSeg(p10,p2));
       thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
 A85:    now assume
A86:      LSeg(p1,p00) /\ LSeg(p10,p2) <> {};
         consider a being Element of LSeg(p1,p00) /\ LSeg(p10,p2);
         reconsider p = a as Point of TOP-REAL 2 by A86,TARSKI:def 3;
           p in LSeg(p00,p1) & p in LSeg(p2,p10) & p00`1 <= p1`1 & p2`1 <= p10
`1
          by A5,A6,A78,A79,A86,EUCLID:56,XBOOLE_0:def 3;
         then p`1 <= p1`1 & p2`1 <= p`1 by TOPREAL1:9;
         hence contradiction by A5,A78,A84,AXIOMS:22;
        end;
  A87:   LSeg(p1,p00) /\ L4 c= {p10} by A7,TOPREAL1:22,XBOOLE_1:26;
          now assume p10 in LSeg(p1,p00) /\ L4;
         then p10 in LSeg(p00,p1) & p00`1 <=
 p1`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         then p10`1 <= p1`1 by TOPREAL1:9;
         hence contradiction by A5,A6,A79,A84,Lm4,AXIOMS:21;
        end;
       then {p10} <> LSeg(p1,p00) /\ L4 by ZFMISC_1:37;
 then A88:   LSeg(p1,p00) /\ L4 = {} by A87,ZFMISC_1:39;
         LSeg(p1,p00) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {}
        by A7,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A89:   LSeg(p1,p00) /\ L2 = {} by XBOOLE_1:3;
A90:
       LSeg(p1,p00) /\ L1 c= L3 /\ L1 by A7,XBOOLE_1:26;
         p00 in LSeg(p1,p00) by TOPREAL1:6;
       then A91: LSeg(p1,p00) /\ L1 <> {} by Lm13,XBOOLE_0:def 3;
 A92:   LSeg(p1,p00) /\ (L1 \/ L2 \/ L4 \/ LSeg(p10,p2))
         = LSeg(p1,p00) /\ (L1 \/ L2 \/ L4) \/ (LSeg(p1,p00) /\ LSeg(p10,p2))
            by XBOOLE_1:23
        .= LSeg(p1,p00) /\ (L1 \/ L2) \/ (LSeg(p1,p00) /\ L4) by A85,XBOOLE_1:
23
        .= LSeg(p1,p00) /\ L1 \/ (LSeg(p1,p00) /\ L2) by A88,XBOOLE_1:23
        .= {p00} by A89,A90,A91,TOPREAL1:23,ZFMISC_1:39;
      L1 is_an_arc_of p00,p01 by Lm4,TOPREAL1:15;
 then A93:   L1 \/ L2 is_an_arc_of p00,p11 by TOPREAL1:16,21;
      (L1 \/ L2) /\ L4 = L1 /\ L4 \/ L2 /\ L4 by XBOOLE_1:23
                     .= {p11} by Lm3,TOPREAL1:24;
 then A94:   L1 \/ L2 \/ L4 is_an_arc_of p00,p10 by A93,TOPREAL1:16;
A95:
       L4 /\ LSeg(p10,p2) c= L4 /\ L3 by A81,XBOOLE_1:26;
         p10 in LSeg(p10,p2) by TOPREAL1:6;
       then L4 /\ LSeg(p10,p2) <> {} by Lm18,XBOOLE_0:def 3;
 then A96:   L4 /\ LSeg(p10,p2) = {p10} by A95,TOPREAL1:22,ZFMISC_1:39;
  A97:   L1 /\ LSeg(p10,p2) c= {p00} by A81,TOPREAL1:23,XBOOLE_1:26;
          now assume p00 in L1 /\ LSeg(p10,p2);
         then p00 in
 LSeg(p2,p10) & p2`1 <= p10`1 by A78,A79,EUCLID:56,XBOOLE_0:def 3;
         hence contradiction by A6,A78,A84,Lm4,TOPREAL1:9;
        end;
       then {p00} <> L1 /\ LSeg(p10,p2) by ZFMISC_1:37;
 then A98:   L1 /\ LSeg(p10,p2) = {} by A97,ZFMISC_1:39;
         L2 /\ LSeg(p10,p2) c= L2 /\ L3 & L3 /\ L2 = {}
        by A81,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A99:   L2 /\ LSeg(p10,p2) = {} by XBOOLE_1:3;
         (L1 \/ L2 \/ L4) /\ LSeg(p10,p2)
         = (L1 \/ L2) /\ LSeg(p10,p2) \/ L4 /\ LSeg(p10,p2) by XBOOLE_1:23
        .= (L1 /\ LSeg(p10,p2)) \/ (L2 /\ LSeg(p10,p2)) \/
 {p10} by A96,XBOOLE_1:23
        .= {p10} by A98,A99;
       then L1 \/ L2 \/ L4 \/
 LSeg(p10,p2) is_an_arc_of p00,p2 by A94,TOPREAL1:16;
       hence P2 is_an_arc_of p1,p2 by A92,TOPREAL1:17;
       thus P1 \/ P2
         = LSeg(p00,p1) \/ LSeg(p1,p2) \/ (L1 \/ L2 \/ L4 \/
 LSeg(p10,p2)) by XBOOLE_1:4
        .= LSeg(p00,p1) \/ LSeg(p1,p2) \/ LSeg(p2,p10) \/ (L1 \/ L2 \/
 L4) by XBOOLE_1:4
        .= (L1 \/ L2 \/ L4) \/ L3 by A3,A77,TOPREAL1:13
        .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
         p1 in LSeg(p1,p2) & p1 in LSeg(p1,p00) by TOPREAL1:6;
       then p1 in LSeg(p1,p2) /\ LSeg(p1,p00) by XBOOLE_0:def 3;
  then A100:   {p1} c= LSeg(p1,p2) /\ LSeg(p1,p00) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p1,p00) c= {p1}
        proof
         let a; assume A101: a in LSeg(p1,p2) /\ LSeg(p1,p00);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p1,p2) & p in LSeg(p00,p1) & p1`2 <= p2`2 & p1`1 <= p2`1
           & p00`1 <= p1`1 by A5,A6,A78,A79,A84,A101,EUCLID:56,XBOOLE_0:def 3;
          then p1`2 <= p`2 & p`2 <= p2`2 & p1`1 <= p`1 & p`1 <= p1`1
           by TOPREAL1:9,10;
          then p1`1 = p`1 & p`2 = 0 by A5,A6,A78,A79,AXIOMS:21;
          then p = |[p1`1, 0]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
          hence a in {p1} by TARSKI:def 1;
        end;
 then A102:   LSeg(p1,p2) /\ LSeg(p1,p00) = {p1} by A100,XBOOLE_0:def 10;
         p2 in LSeg(p1,p2) & p2 in LSeg(p10,p2) by TOPREAL1:6;
       then p2 in LSeg(p1,p2) /\ LSeg(p10,p2) by XBOOLE_0:def 3;
  then A103:   {p2} c= LSeg(p1,p2) /\ LSeg(p10,p2) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p10,p2) c= {p2}
        proof
         let a; assume A104: a in LSeg(p1,p2) /\ LSeg(p10,p2);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p1,p2) & p in LSeg(p2,p10) & p1`2 <= p2`2 & p1`1 <= p2`1
           & p2`1 <= p10`1 by A5,A6,A78,A79,A84,A104,EUCLID:56,XBOOLE_0:def 3;
          then p1`2 <= p`2 & p`2 <= p2`2 & p2`1 <= p`1 & p`1 <= p2`1
           by TOPREAL1:9,10;
          then p2`1 = p`1 & p`2 = 0 by A5,A6,A78,A79,AXIOMS:21;
          then p = |[ p2`1, 0]| by EUCLID:57 .= p2 by A78,A79,EUCLID:57;
          hence a in {p2} by TARSKI:def 1;
        end;
 then A105:   LSeg(p1,p2) /\ LSeg(p10,p2) = {p2} by A103,XBOOLE_0:def 10;
         LSeg(p1,p2) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {}
        by A83,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A106:   LSeg(p1,p2) /\ L2 = {} by XBOOLE_1:3;
  A107:   P1 /\ P2
         = (LSeg(p1,p2) /\ LSeg(p1,p00))
           \/ (LSeg(p1,p2) /\ (L1 \/ L2 \/ L4 \/ LSeg(p10,p2))) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L2 \/ L4)) \/
 {p2}) by A102,A105,XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L2)) \/ (LSeg(p1,p2) /\ L4) \/ {p2})
            by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ (LSeg(p1,p2) /\ L2)
            \/ (LSeg(p1,p2) /\ L4) \/ {p2}) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2}))
            by A106,XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2})
            by XBOOLE_1:4;
  A108:    now per cases;
         suppose A109: p1 = p00;
A110:          LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A83,XBOOLE_1:26;
            p1 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L1 <> {} by A109,Lm13,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L1 = {p1} by A109,A110,TOPREAL1:23,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A107;
         suppose A111: p1 <> p00;
A112:          LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A83,XBOOLE_1:26;
             now assume p00 in LSeg(p1,p2) /\ L1;
            then p00 in LSeg(p1,p2) & p1`1 <= p2`1
             by A5,A78,A84,XBOOLE_0:def 3;
            then p1`1 = 0 by A5,A6,Lm4,TOPREAL1:9;
            hence contradiction by A5,A6,A111,EUCLID:57;
           end;
          then {p00} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L1 = {} by A112,TOPREAL1:23,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A107;
        end;
          now per cases;
         suppose A113: p2 = p10;
  then A114:      LSeg(p1,p2) /\ L4 c= {p2} by A83,TOPREAL1:22,XBOOLE_1:26;
            p2 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L4 <> {} by A113,Lm18,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L4 = {p2} by A114,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A108,ENUMSET1:41;
         suppose A115: p2 <> p10;
  A116:      LSeg(p1,p2) /\ L4 c= {p10} by A83,TOPREAL1:22,XBOOLE_1:26;
             now assume p10 in LSeg(p1,p2) /\ L4;
            then p10 in LSeg(p1,p2) & p1`1 <= p2`1
             by A5,A78,A84,XBOOLE_0:def 3;
            then p10`1 <= p2`1 by TOPREAL1:9;
            then p2`1 = 1 by A78,A79,Lm4,AXIOMS:21;
            hence contradiction by A78,A79,A115,EUCLID:57;
           end;
          then {p10} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L4 = {} by A116,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A108,ENUMSET1:41;
        end;
        hence P1 /\ P2 = {p1,p2};
       suppose A117: q`1 < p`1;
       take P1 = LSeg(p1,p2),P2 = LSeg(p1,p10) \/ (L4 \/ L2 \/ L1 \/
 LSeg(p00,p2));
       thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
 A118:    now assume
A119:      LSeg(p1,p10) /\ LSeg(p00,p2) <> {};
         consider a being Element of LSeg(p1,p10) /\ LSeg(p00,p2);
         reconsider p = a as Point of TOP-REAL 2 by A119,TARSKI:def 3;
           p in LSeg(p1,p10) & p in LSeg(p00,p2) & p1`1 <= p10`1 & p00`1 <= p2
`1
          by A5,A6,A78,A79,A119,EUCLID:56,XBOOLE_0:def 3;
         then p1`1 <= p`1 & p`1 <= p2`1 by TOPREAL1:9;
         hence contradiction by A5,A78,A117,AXIOMS:22;
        end;
A120:
       LSeg(p1,p10) /\ L1 c= L3 /\ L1 by A8,XBOOLE_1:26;
          now assume p00 in LSeg(p1,p10) /\ L1;
         then p00 in LSeg(p1,p10) & p1`1 <=
 p10`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         hence contradiction by A5,A79,A117,Lm4,TOPREAL1:9;
        end;
       then {p00} <> LSeg(p1,p10) /\ L1 by ZFMISC_1:37;
 then A121:   LSeg(p1,p10) /\ L1 = {} by A120,TOPREAL1:23,ZFMISC_1:39;
         LSeg(p1,p10) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {}
        by A8,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A122:   LSeg(p1,p10) /\ L2 = {} by XBOOLE_1:3;
  A123:   LSeg(p1,p10) /\ L4 c= {p10} by A8,TOPREAL1:22,XBOOLE_1:26;
         p10 in LSeg(p1,p10) by TOPREAL1:6;
       then A124: LSeg(p1,p10) /\ L4 <> {} by Lm18,XBOOLE_0:def 3;
 A125:   LSeg(p1,p10) /\ (L4 \/ L2 \/ L1 \/ LSeg(p00,p2))
         = LSeg(p1,p10) /\ (L4 \/ L2 \/ L1) \/ (LSeg(p1,p10) /\ LSeg(p00,p2))
            by XBOOLE_1:23
        .= LSeg(p1,p10) /\ (L4 \/ L2) \/ (LSeg(p1,p10) /\ L1) by A118,XBOOLE_1:
23
        .= LSeg(p1,p10) /\ L4 \/ (LSeg(p1,p10) /\ L2) by A121,XBOOLE_1:23
        .= {p10} by A122,A123,A124,ZFMISC_1:39;
      L4 is_an_arc_of p10,p11 by Lm4,TOPREAL1:15;
 then A126:   L4 \/ L2 is_an_arc_of p10,p01 by TOPREAL1:16,24;
      (L4 \/ L2) /\ L1 = L1 /\ L4 \/ L2 /\ L1 by XBOOLE_1:23
                     .= {p01} by Lm3,TOPREAL1:21;
 then A127:   L4 \/ L2 \/ L1 is_an_arc_of p10,p00 by A126,TOPREAL1:16;
  A128:   L1 /\ LSeg(p00,p2) c= {p00} by A82,TOPREAL1:23,XBOOLE_1:26;
         p00 in LSeg(p00,p2) by TOPREAL1:6;
       then L1 /\ LSeg(p00,p2) <> {} by Lm13,XBOOLE_0:def 3;
 then A129:   L1 /\ LSeg(p00,p2) = {p00} by A128,ZFMISC_1:39;
A130:
       L4 /\ LSeg(p00,p2) c= L4 /\ L3 by A82,XBOOLE_1:26;
          now assume p10 in L4 /\ LSeg(p00,p2);
         then p10 in
 LSeg(p00,p2) & p00`1 <= p2`1 by A78,A79,EUCLID:56,XBOOLE_0:def 3;
         then p10`1 <= p2`1 by TOPREAL1:9;
         hence contradiction by A6,A78,A79,A117,Lm4,AXIOMS:21;
        end;
       then {p10} <> L4 /\ LSeg(p00,p2) by ZFMISC_1:37;
 then A131:   L4 /\ LSeg(p00,p2) = {} by A130,TOPREAL1:22,ZFMISC_1:39;
         L2 /\ LSeg(p00,p2) c= L2 /\ L3 & L3 /\ L2 = {}
        by A82,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A132:   L2 /\ LSeg(p00,p2) = {} by XBOOLE_1:3;
         (L4 \/ L2 \/ L1) /\ LSeg(p00,p2)
         = (L4 \/ L2) /\ LSeg(p00,p2) \/ L1 /\ LSeg(p00,p2) by XBOOLE_1:23
        .= (L4 /\ LSeg(p00,p2)) \/ (L2 /\ LSeg(p00,p2)) \/
 {p00} by A129,XBOOLE_1:23
        .= {p00} by A131,A132;
       then L4 \/ L2 \/ L1 \/
 LSeg(p00,p2) is_an_arc_of p10,p2 by A127,TOPREAL1:16
;
       hence P2 is_an_arc_of p1,p2 by A125,TOPREAL1:17;
       thus P1 \/ P2
         = LSeg(p2,p1) \/ LSeg(p1,p10) \/ (L4 \/ L2 \/ L1 \/
 LSeg(p00,p2)) by XBOOLE_1:4
        .= LSeg(p00,p2) \/ (LSeg(p2,p1) \/ LSeg(p1,p10)) \/ (L4 \/ L2 \/ L1)
            by XBOOLE_1:4
        .= L3 \/ (L4 \/ L2 \/ L1) by A3,A77,TOPREAL1:13
        .= L3 \/ (L4 \/ (L1 \/ L2)) by XBOOLE_1:4
        .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
          p1 in LSeg(p1,p2) & p1 in LSeg(p1,p10) by TOPREAL1:6;
        then p1 in LSeg(p1,p2) /\ LSeg(p1,p10) by XBOOLE_0:def 3;
  then A133:    {p1} c= LSeg(p1,p2) /\ LSeg(p1,p10) by ZFMISC_1:37;
          LSeg(p1,p2) /\ LSeg(p1,p10) c= {p1}
         proof
          let a; assume A134: a in LSeg(p1,p2) /\ LSeg(p1,p10);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p2,p1) & p in LSeg(p1,p10) & p2`2 <= p1`2 & p2`1 <= p1`1
           & p1`1 <= p10`1 by A5,A6,A78,A79,A117,A134,EUCLID:56,XBOOLE_0:def 3;
          then p2`2 <= p`2 & p`2 <= p1`2 & p1`1 <= p`1 & p`1 <= p1`1
           by TOPREAL1:9,10;
          then p1`1 = p`1 & p`2 = 0 by A5,A6,A78,A79,AXIOMS:21;
          then p = |[p1`1, 0]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
          hence a in {p1} by TARSKI:def 1;
         end;
 then A135:    LSeg(p1,p2) /\ LSeg(p1,p10) = {p1} by A133,XBOOLE_0:def 10;
          p2 in LSeg(p1,p2) & p2 in LSeg(p00,p2) by TOPREAL1:6;
        then p2 in LSeg(p1,p2) /\ LSeg(p00,p2) by XBOOLE_0:def 3;
  then A136:    {p2} c= LSeg(p1,p2) /\ LSeg(p00,p2) by ZFMISC_1:37;
          LSeg(p1,p2) /\ LSeg(p00,p2) c= {p2}
         proof
          let a; assume A137: a in LSeg(p1,p2) /\ LSeg(p00,p2);
          then reconsider p = a as Point of TOP-REAL 2;
            p in LSeg(p2,p1) & p in LSeg(p00,p2) & p2`1 <= p1`1 & p2`2 <= p1`2
           & p00`1 <= p2`1 by A5,A6,A78,A79,A117,A137,EUCLID:56,XBOOLE_0:def 3;
          then p2`2 <= p`2 & p`2 <= p1`2 & p2`1 <= p`1 & p`1 <= p2`1
           by TOPREAL1:9,10;
          then p2`1 = p`1 & p`2 = 0 by A5,A6,A78,A79,AXIOMS:21;
          then p = |[ p2`1, 0]| by EUCLID:57 .= p2 by A78,A79,EUCLID:57;
          hence a in {p2} by TARSKI:def 1;
         end;
 then A138:    LSeg(p1,p2) /\ LSeg(p00,p2) = {p2} by A136,XBOOLE_0:def 10;
          LSeg(p1,p2) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {}
         by A83,TOPREAL1:25,XBOOLE_0:def 7,XBOOLE_1:26;
 then A139:    LSeg(p1,p2) /\ L2 = {} by XBOOLE_1:3;
  A140:    P1 /\ P2
         = {p1} \/ LSeg(p1,p2) /\ (L4 \/ L2 \/ L1 \/
 LSeg(p00,p2)) by A135,XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L2 \/ L1)) \/ {p2}) by A138,XBOOLE_1
:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L2)) \/ (LSeg(p1,p2) /\ L1) \/ {p2})
            by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ (LSeg(p1,p2) /\ L2)
            \/ (LSeg(p1,p2) /\ L1) \/ {p2}) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/ {p2}))
            by A139,XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/
 {p2}) by XBOOLE_1:4;
  A141:    now per cases;
         suppose A142: p2 = p00;
A143:          LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A83,XBOOLE_1:26;
            p2 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L1 <> {} by A142,Lm13,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L1 = {p2} by A142,A143,TOPREAL1:23,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A140;
         suppose A144: p2 <> p00;
A145:          LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A83,XBOOLE_1:26;
             now assume p00 in LSeg(p1,p2) /\ L1;
            then p00 in LSeg(p2,p1) & p2`1 <= p1`1
             by A5,A78,A117,XBOOLE_0:def 3;
            then p2`1 = 0 by A78,A79,Lm4,TOPREAL1:9;
            hence contradiction by A78,A79,A144,EUCLID:57;
           end;
          then {p00} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L1 = {} by A145,TOPREAL1:23,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A140;
        end;
          now per cases;
         suppose A146: p1 = p10;
  then A147:      LSeg(p1,p2) /\ L4 c= {p1} by A83,TOPREAL1:22,XBOOLE_1:26;
            p1 in LSeg(p1,p2) by TOPREAL1:6;
          then LSeg(p1,p2) /\ L4 <> {} by A146,Lm18,XBOOLE_0:def 3;
          then LSeg(p1,p2) /\ L4 = {p1} by A147,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A141,ENUMSET1:41;
         suppose A148: p1 <> p10;
  A149:      LSeg(p1,p2) /\ L4 c= {p10} by A83,TOPREAL1:22,XBOOLE_1:26;
             now assume p10 in LSeg(p1,p2) /\ L4;
            then p10 in LSeg(p2,p1) & p2`1 <= p1`1
             by A5,A78,A117,XBOOLE_0:def 3;
            then p10`1 <= p1`1 by TOPREAL1:9;
            then p1`1 = 1 by A5,A6,Lm4,AXIOMS:21;
            hence contradiction by A5,A6,A148,EUCLID:57;
           end;
          then {p10} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:37;
          then LSeg(p1,p2) /\ L4 = {} by A149,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A141,ENUMSET1:41;
        end;
       hence P1 /\ P2 = {p1,p2};
     end;
    hence ex P1,P2 being non empty Subset of TOP-REAL 2
    st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
     R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2};
    suppose A150: p2 in L4;
     then A151: ex q st q = p2 & q`1 = 1 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:19;
 A152: LSeg(p2,p10) c= L4 by A150,Lm18,TOPREAL1:12;
 A153: LSeg(p2,p11) c= L4 by A150,Lm20,TOPREAL1:12;
     take P1 = LSeg(p1,p10) \/ LSeg(p10,p2),P2
 = LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2));
       p10 in LSeg(p1,p10) & p10 in LSeg(p10,p2) by TOPREAL1:6;
     then LSeg(p1,p10) /\ LSeg(p10,p2) c= L3 /\ L4 & L3 /\ L4 = {p10}
      & LSeg(p1,p10) /\ LSeg(p10,p2) <> {} by A8,A152,TOPREAL1:22,XBOOLE_0:def
3,XBOOLE_1:27;
     then LSeg(p1,p10) /\ LSeg(p10,p2) = {p10} & (p1 <> p10 or p2 <> p10)
      by A1,ZFMISC_1:39;
     hence P1 is_an_arc_of p1,p2 by TOPREAL1:18;
    L1 is_an_arc_of p00,p01 & L2 is_an_arc_of p01,p11 by Lm4,TOPREAL1:15;
 then A154: L1 \/ L2 is_an_arc_of p00,p11 by TOPREAL1:5,21;
       L1 /\ LSeg(p11,p2) c= L1 /\ L4 by A153,XBOOLE_1:26;
 then A155: L1 /\ LSeg(p11,p2) = {} by Lm3,XBOOLE_1:3;
       p11 in LSeg(p11,p2) by TOPREAL1:6;
     then A156: L2 /\ LSeg(p11,p2) c= {p11} & L2 /\ LSeg(p11,p2) <> {}
      by A153,Lm19,TOPREAL1:24,XBOOLE_0:def 3,XBOOLE_1:27;
       (L1 \/ L2) /\ LSeg(p11,p2)
       = (L1 /\ LSeg(p11,p2)) \/ (L2 /\ LSeg(p11,p2)) by XBOOLE_1:23
      .= {p11} by A155,A156,ZFMISC_1:39;
 then A157: L1 \/ L2 \/ LSeg(p11,p2) is_an_arc_of p00,p2 by A154,TOPREAL1:16;
     A158: LSeg(p1,p00) /\
 LSeg(p11,p2) c= {p10} by A7,A153,TOPREAL1:22,XBOOLE_1:27
;
       now assume p10 in LSeg(p1,p00) /\ LSeg(p11,p2);
      then p10 in LSeg(p00,p1) & p10 in LSeg(p2,p11) & p00`1 <= p1`1 &
       p2`2 <= p11`2 by A5,A6,A151,EUCLID:56,XBOOLE_0:def 3;
      then p10`1 <= p1`1 & p2`2 <= p10`2 by TOPREAL1:9,10;
      then 0 = p1`2 & 0 = p2`2 & 1 = p1`1 & 1 = p2`1
       by A5,A6,A151,Lm4,AXIOMS:21;
      then p1 = p10 & p2 = p10 by EUCLID:57;
      hence contradiction by A1;
     end;
     then {p10} <> LSeg(p1,p00) /\ LSeg(p11,p2) by ZFMISC_1:37;
 then A159: LSeg(p1,p00) /\ LSeg(p11,p2) = {} by A158,ZFMISC_1:39;
       LSeg(p1,p00) /\ (L1 \/ L2 \/ LSeg(p11,p2))
       = (LSeg(p1,p00) /\ (L1 \/ L2)) \/ (LSeg(p1,p00) /\
 LSeg(p11,p2)) by XBOOLE_1:23
      .= (LSeg(p1,p00) /\ L1) \/ (LSeg(p1,p00) /\ L2) by A159,XBOOLE_1:23
      .= {p00} by A11,A13,TOPREAL1:23,ZFMISC_1:39;
     hence P2 is_an_arc_of p1,p2 by A157,TOPREAL1:17;
A160:  L3 = LSeg(p1,p10) \/ LSeg(p1,p00) by A3,TOPREAL1:11;
A161:  L4 = LSeg(p11,p2) \/ LSeg(p10,p2) by A150,TOPREAL1:11;
     thus P1 \/ P2
      = LSeg(p10,p2) \/ (LSeg(p1,p10) \/ (LSeg(p1,p00) \/
         (L1 \/ L2 \/ LSeg(p11,p2)))) by XBOOLE_1:4
     .= LSeg(p10,p2) \/ (L3 \/ (L1 \/ L2 \/ LSeg(p11,p2))) by A160,XBOOLE_1:4
     .= (L1 \/ L2) \/ L3 \/ LSeg(p11,p2) \/ LSeg(p10,p2) by XBOOLE_1:4
     .= (L1 \/ L2) \/ L3 \/ L4 by A161,XBOOLE_1:4
     .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
A162:  LSeg(p1,p10) /\ LSeg(p1,p00)
       = {p1} by A3,TOPREAL1:14;
A163:  LSeg(p10,p2) /\ LSeg(p11,p2) = {p2} by A150,TOPREAL1:14;
       LSeg(p10,p2) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
      by A152,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
then A164:  LSeg(p10,p2) /\ L1 = {} by XBOOLE_1:3;
A165:   P1 /\ P2
      = (LSeg(p1,p10) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2))))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/
 LSeg(p11,p2)))) by XBOOLE_1:23
     .= (LSeg(p1,p10) /\ LSeg(p1,p00))
      \/ (LSeg(p1,p10) /\ (L1 \/ L2 \/ LSeg(p11,p2)))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/
 LSeg(p11,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ (L1 \/ L2))
      \/ (LSeg(p1,p10) /\ LSeg(p11,p2)))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2))))
       by A162,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p10,p1) /\ L2)
      \/ (LSeg(p1,p10) /\ LSeg(p11,p2)))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/
 LSeg(p11,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p00))
      \/ (LSeg(p10,p2) /\ (L1 \/ L2 \/ LSeg(p11,p2)))) by A9,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p00))
      \/ ((LSeg(p10,p2) /\ (L1 \/ L2)) \/ {p2})) by A163,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p00))
      \/ ((LSeg(p10,p2) /\ L1) \/ (LSeg(p10,p2) /\ L2) \/ {p2})) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p00))
      \/ ((LSeg(p10,p2) /\ L2) \/ {p2})) by A164;
A166:   now per cases;
      suppose A167: p1 = p00;
 then A168:   LSeg(p10,p2) /\ LSeg(p1,p00) = LSeg(p10,p2) /\
 {p00} by TOPREAL1:7;
        not p00 in LSeg(p10,p2) by A152,Lm4,TOPREAL1:9;
       then LSeg(p10,p2) /\ LSeg(p1,p00) = {} by A168,Lm1;
     hence P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2))
         \/ ((LSeg(p10,p2) /\ L2) \/
 {p2}) by A165,A167,TOPREAL1:23,XBOOLE_1:4
       .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2))
         \/ ((LSeg(p10,p2) /\ L2) \/ {p2});
      suppose A169: p1 = p10;
    then LSeg(p1,p10) /\ L1 = {p10} /\ L1 by TOPREAL1:7;
then A170:   LSeg(p1,p10) /\ L1 = {} by Lm1,Lm9;
A171:      LSeg(p10,p2) /\ LSeg(p1,p00) c= L4 /\ L3 by A7,A152,XBOOLE_1:27;
        p1 in LSeg(p10,p2) & p1 in LSeg(p1,p00) by A169,TOPREAL1:6;
      then {} <> LSeg(p10,p2) /\ LSeg(p1,p00) by XBOOLE_0:def 3;
   then LSeg(p10,p2) /\ LSeg(p1,p00) = {p1} by A169,A171,TOPREAL1:22,ZFMISC_1:
39;
      hence P1 /\ P2 = {p1} \/ ({p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2)))
        \/ ((LSeg(p10,p2) /\ L2) \/ {p2}) by A165,A170,XBOOLE_1:4
        .= {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2))
        \/ ((LSeg(p10,p2) /\ L2) \/ {p2}) by XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2))
        \/ ((LSeg(p10,p2) /\ L2) \/ {p2});
      suppose A172: p1 <> p10 & p1 <> p00;
A173:
      LSeg(p1,p10) /\ L1 c= L3 /\ L1 by A8,XBOOLE_1:26;
         now assume p00 in LSeg(p1,p10) /\ L1;
        then p00 in
 LSeg(p1,p10) & p1`1 <= p10`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
        then p1`1 = 0 by A5,A6,Lm4,TOPREAL1:9;
        hence contradiction by A5,A6,A172,EUCLID:57;
       end;
      then {p00} <> LSeg(p1,p10) /\ L1 by ZFMISC_1:37;
then A174:   LSeg(p1,p10) /\ L1 = {} by A173,TOPREAL1:23,ZFMISC_1:39;
A175:      LSeg(p10,p2) /\ LSeg(p1,p00) c= L4 /\ L3 by A7,A152,XBOOLE_1:27;
         now assume p10 in LSeg(p10,p2) /\ LSeg(p1,p00);
        then p10 in
 LSeg(p00,p1) & p00`1 <= p1`1 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
        then p10`1 <= p1`1 by TOPREAL1:9;
        then p1`1 = 1 by A5,A6,Lm4,AXIOMS:21;
        hence contradiction by A5,A6,A172,EUCLID:57;
       end;
      then {p10} <> LSeg(p10,p2) /\ LSeg(p1,p00) by ZFMISC_1:37;
   then LSeg(p10,p2) /\ LSeg(p1,p00) = {} by A175,TOPREAL1:22,ZFMISC_1:39;
      hence P1 /\ P2 = {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2))
        \/ ((LSeg(p10,p2) /\ L2) \/ {p2}) by A165,A174;
     end;
       now per cases;
      suppose A176: p2 = p10;
 then A177:   LSeg(p1,p10) /\ LSeg(p11,p2) c= {p2} by A8,TOPREAL1:22,XBOOLE_1:
27;
        p10 in LSeg(p1,p10) by TOPREAL1:6;
      then {} <> LSeg(p1,p10) /\ LSeg(p11,p2) by A176,Lm18,XBOOLE_0:def 3;
then A178:   LSeg(p1,p10) /\ LSeg(p11,p2) = {p2} by A177,ZFMISC_1:39;
      LSeg(p10,p2) /\ L2 = {p10} /\ L2 by A176,TOPREAL1:7;
   then LSeg(p10,p2) /\ L2 = {} by Lm1,Lm10;
      hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A166,A178,XBOOLE_1:4
        .= {p1,p2} by ENUMSET1:41;
      suppose A179: p2 = p11;
 then A180:   LSeg(p1,p10) /\ LSeg(p11,p2) = LSeg(p1,p10) /\ {p11} by TOPREAL1:
7;
        not p11 in LSeg(p1,p10) by A8,Lm4,TOPREAL1:10;
      then LSeg(p1,p10) /\ LSeg(p11,p2) = {} by A180,Lm1;
      hence P1 /\ P2 = {p1,p2} by A166,A179,ENUMSET1:41,TOPREAL1:24;
      suppose A181: p2 <> p11 & p2 <> p10;
 A182:   LSeg(p1,p10) /\ LSeg(p11,p2) c= {p10} by A8,A153,TOPREAL1:22,XBOOLE_1:
27;
         now assume p10 in LSeg(p1,p10) /\ LSeg(p11,p2);
        then p10 in LSeg(p2,p11) & p2`2 <= p11`2 by A151,EUCLID:56,XBOOLE_0:def
3;
        then p2`2 = 0 by A151,Lm4,TOPREAL1:10;
        hence contradiction by A151,A181,EUCLID:57;
       end;
      then {p10} <> LSeg(p1,p10) /\ LSeg(p11,p2) by ZFMISC_1:37;
then A183:   LSeg(p1,p10) /\ LSeg(p11,p2) = {} by A182,ZFMISC_1:39;
A184:
      LSeg(p10,p2) /\ L2 c= L4 /\ L2 by A152,XBOOLE_1:26;
         now assume p11 in LSeg(p10,p2) /\ L2;
        then p11 in LSeg(p10,p2) & p10`2 <= p2`2 by A151,EUCLID:56,XBOOLE_0:def
3;
        then p11`2 <= p2`2 by TOPREAL1:10;
        then 1 = p2`2 by A151,Lm4,AXIOMS:21;
        hence contradiction by A151,A181,EUCLID:57;
       end;
      then {p11} <> LSeg(p10,p2) /\ L2 by ZFMISC_1:37;
   then LSeg(p10,p2) /\ L2 = {} by A184,TOPREAL1:24,ZFMISC_1:39;
      hence P1 /\ P2 = {p1,p2} by A166,A183,ENUMSET1:41;
     end;
   hence P1 /\ P2 = {p1,p2};
 end;

Lm26:
 p1 <> p2 & p2 in R^2-unit_square &
 p1 in LSeg(p10, p11) implies
  ex P1,P2 being non empty Subset of TOP-REAL 2
  st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
   R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2}
 proof assume that
 A1: p1 <> p2 and
 A2: p2 in R^2-unit_square and
 A3: p1 in LSeg(p10, p11);
A4:   p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:20,XBOOLE_0:def 2;
   consider p such that
A5: p = p1 and A6: p`1 = 1 & p`2 <= 1 & p`2 >= 0 by A3,TOPREAL1:19;
A7: LSeg(p1,p11) c= L4 by A3,Lm20,TOPREAL1:12;
A8: LSeg(p1,p10) c= L4 by A3,Lm18,TOPREAL1:12;
    then A9:
    LSeg(p1,p10) /\ L3 c= L4 /\ L3 by XBOOLE_1:26;
      p10 in LSeg(p1,p10) by TOPREAL1:6;
    then A10: {} <> LSeg(p1,p10) /\ L3 by Lm17,XBOOLE_0:def 3;
A11:
    LSeg(p1,p11) /\ L2 c= L4 /\ L2 by A7,XBOOLE_1:26;
      p11 in LSeg(p1,p11) by TOPREAL1:6;
    then A12: {} <> LSeg(p1,p11) /\ L2 by Lm19,XBOOLE_0:def 3;
      LSeg(p10,p1) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
      by A8,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
then A13: LSeg(p10,p1) /\ L1 = {} by XBOOLE_1:3;
      LSeg(p1,p11) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
     by A7,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
then A14: LSeg(p1,p11) /\ L1 = {} by XBOOLE_1:3;
   per cases by A4,XBOOLE_0:def 2;
    suppose A15: p2 in L1;
    then A16: ex q st q = p2 & q`1 = 0 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:19;
    take P1 = LSeg(p1,p11) \/ L2 \/ LSeg(p01,p2),P2
 = LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2);
 A17: LSeg(p2,p01) c= L1 by A15,Lm15,TOPREAL1:12;
 A18: LSeg(p2,p00) c= L1 by A15,Lm13,TOPREAL1:12;
       p01 in LSeg(p01,p2) by TOPREAL1:6;
     then L2 /\ LSeg(p01,p2) c= L2 /\ L1 &
      p01 in L2 /\ LSeg(p01,p2) by A17,Lm16,XBOOLE_0:def 3,XBOOLE_1:26;
 then A19: L2 /\ LSeg(p01,p2) = {p01} by TOPREAL1:21,ZFMISC_1:39;
       L2 is_an_arc_of p11,p01 by Lm4,TOPREAL1:15;
     then A20: L2 \/ LSeg(p01,p2) is_an_arc_of p11,p2 by A19,TOPREAL1:16;
       LSeg(p1,p11) /\ LSeg(p01,p2) c= L4 /\ L1 by A7,A17,XBOOLE_1:27;
  then A21: LSeg(p1,p11) /\ LSeg(p01,p2) = {} by Lm3,XBOOLE_1:3;
       LSeg(p1,p11) /\ (L2 \/ LSeg(p01,p2))
       = (LSeg(p1,p11) /\ L2) \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) by XBOOLE_1:23
      .= {p11} by A11,A12,A21,TOPREAL1:24,ZFMISC_1:39;
     then LSeg(p1,p11) \/ (L2 \/ LSeg(p01,p2)) is_an_arc_of p1,p2
      by A20,TOPREAL1:17;
     hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4;
       L3 /\ LSeg(p00,p2) c= L3 /\ L1 &
      p00 in LSeg(p00,p2) by A18,TOPREAL1:6,XBOOLE_1:26;
     then L3 /\ LSeg(p00,p2) c= {p00} & L3 /\ LSeg(p00,p2) <> {}
      by Lm14,TOPREAL1:23,XBOOLE_0:def 3;
 then A22: L3 /\ LSeg(p00,p2) = {p00} by ZFMISC_1:39;
       L3 is_an_arc_of p10,p00 by Lm4,TOPREAL1:15;
 then A23: L3 \/ LSeg(p00,p2) is_an_arc_of p10,p2 by A22,TOPREAL1:16;
       LSeg(p1,p10) /\ LSeg(p00,p2) c= L4 /\ L1 by A8,A18,XBOOLE_1:27;
 then A24: LSeg(p1,p10) /\ LSeg(p00,p2) = {} by Lm3,XBOOLE_1:3;
       LSeg(p1,p10) /\ (L3 \/ LSeg(p00,p2))
       = (LSeg(p1,p10) /\ L3) \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) by XBOOLE_1:23
      .= {p10} by A9,A10,A24,TOPREAL1:22,ZFMISC_1:39;
     then LSeg(p1,p10) \/ (L3 \/ LSeg(p00,p2)) is_an_arc_of p1,p2
      by A23,TOPREAL1:17;
    hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4;
A25: LSeg(p1,p11) \/ LSeg(p1,p10) = L4 by A3,TOPREAL1:11;
    thus R^2-unit_square = LSeg(p00,p2) \/ LSeg(p01,p2) \/ L2 \/ (L3 \/
 L4) by A15,TOPREAL1:11,20
      .= LSeg(p00,p2) \/ (LSeg(p01,p2) \/ L2) \/ (L3 \/ L4) by XBOOLE_1:4
      .= L2 \/ LSeg(p01,p2) \/ (L4 \/ L3 \/ LSeg(p00,p2)) by XBOOLE_1:4
      .= L2 \/ LSeg(p01,p2) \/ (L4 \/ (L3 \/ LSeg(p00,p2))) by XBOOLE_1:4
      .= L2 \/ LSeg(p01,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p10)
          \/ (L3 \/ LSeg(p00,p2)))) by A25,XBOOLE_1:4
      .= L2 \/ LSeg(p01,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p10)
          \/ L3 \/ LSeg(p00,p2))) by XBOOLE_1:4
      .= (LSeg(p1,p11) \/ (L2 \/ LSeg(p01,p2))) \/ (LSeg(p1,p10)
          \/ L3 \/ LSeg(p00,p2)) by XBOOLE_1:4
      .= P1 \/ P2 by XBOOLE_1:4;
A26: LSeg(p01,p2) /\ LSeg(p00,p2) = {p2} by A15,TOPREAL1:14;
      LSeg(p01,p2) /\ LSeg(p1,p10) c= L1 /\ L4 & L1 /\ L4 = {}
     by A8,A17,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:27;
then A27: LSeg(p01,p2) /\ LSeg(p1,p10) = {} by XBOOLE_1:3;
      LSeg(p1,p11) /\ LSeg(p00,p2) c= L4 /\ L1 & L1 /\ L4 = {}
     by A7,A18,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:27;
then A28: LSeg(p1,p11) /\ LSeg(p00,p2) = {} by XBOOLE_1:3;
A29: LSeg(p1,p11) /\ LSeg(p1,p10) = {p1} by A3,TOPREAL1:14;
A30:  P1 /\ P2
     = ((LSeg(p1,p11) \/ L2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)))
       \/ (LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))) by XBOOLE_1:23
    .= ((LSeg(p1,p11) \/ L2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)))
       \/ ((LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L3)) \/ {p2}) by A26,XBOOLE_1:23
    .= ((LSeg(p1,p11) \/ L2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)))
       \/ ((LSeg(p01,p2) /\ LSeg(p1,p10)) \/ (LSeg(p01,p2) /\ L3) \/ {p2})
        by XBOOLE_1:23
    .= ((LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)))
       \/ (L2 /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))))
       \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A27,XBOOLE_1:23
    .= (LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)))
       \/ ((L2 /\ (LSeg(p1,p10) \/ L3)) \/ (L2 /\ LSeg(p00,p2)))
       \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by XBOOLE_1:23
    .= (LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)))
       \/ ((L2 /\ LSeg(p1,p10)) \/ (L3 /\ L2) \/ (L2 /\ LSeg(p00,p2)))
       \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by XBOOLE_1:23
    .= ((LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3)) \/ (LSeg(p1,p11) /\
 LSeg(p00,p2)))
       \/ ((L2 /\ LSeg(p1,p10)) \/ (L2 /\ LSeg(p00,p2)))
       \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by Lm2,XBOOLE_1:23
    .= {p1} \/ (LSeg(p1,p11) /\ L3)
       \/ ((L2 /\ LSeg(p1,p10)) \/ (L2 /\ LSeg(p00,p2)))
       \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A28,A29,XBOOLE_1:23;
A31:   now per cases;
      suppose A32: p1 = p10;
    then L2 /\ LSeg(p1,p10) = L2 /\ {p10} by TOPREAL1:7;
   then L2 /\ LSeg(p1,p10) = {} by Lm1,Lm10;
      hence P1 /\ P2 = {p1} \/ (L2 /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3)
\/
 {p2})
 by A30,A32,TOPREAL1:22;
      suppose A33: p1 = p11;
    then LSeg(p1,p11) /\ L3 = {p11} /\ L3 by TOPREAL1:7;
then LSeg(p1,p11) /\ L3 = {} by Lm1,Lm12;
      hence P1 /\ P2 = {p1} \/ {p1} \/ (L2 /\ LSeg(p00,p2))
         \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A30,A33,TOPREAL1:24,XBOOLE_1:4
        .= {p1} \/ (L2 /\ LSeg(p00,p2))
         \/ ((LSeg(p01,p2) /\ L3) \/ {p2});
      suppose A34: p1 <> p11 & p1 <> p10;
A35:
      LSeg(p1,p11) /\ L3 c= L4 /\ L3 by A7,XBOOLE_1:26;
         now assume p10 in LSeg(p1,p11) /\ L3;
        then p10 in
 LSeg(p1,p11) & p1`2 <= p11`2 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
        then p1`2 = 0 by A5,A6,Lm4,TOPREAL1:10;
        hence contradiction by A5,A6,A34,EUCLID:57;
       end;
      then {p10} <> LSeg(p1,p11) /\ L3 by ZFMISC_1:37;
then A36:   LSeg(p1,p11) /\ L3 = {} by A35,TOPREAL1:22,ZFMISC_1:39;
 A37:   L2 /\ LSeg(p1,p10) c= {p11} by A8,TOPREAL1:24,XBOOLE_1:26;
         now assume p11 in L2 /\ LSeg(p1,p10);
        then p11 in
 LSeg(p10,p1) & p10`2 <= p1`2 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
        then p11`2 <= p1`2 by TOPREAL1:10;
        then 1 = p1`2 by A5,A6,Lm4,AXIOMS:21;
        hence contradiction by A5,A6,A34,EUCLID:57;
       end;
      then {p11} <> L2 /\ LSeg(p1,p10) by ZFMISC_1:37;
   then L2 /\ LSeg(p1,p10) = {} by A37,ZFMISC_1:39;
      hence P1 /\ P2 = {p1} \/ (L2 /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3)
\/
 {p2})
 by A30,A36;
     end;
      now per cases;
     suppose A38: p2 = p00;
   then L2 /\ LSeg(p00,p2) = L2 /\ {p00} by TOPREAL1:7;
then L2 /\ LSeg(p00,p2) = {} by Lm1,Lm6;
     hence P1 /\ P2 = {p1,p2} by A31,A38,ENUMSET1:41,TOPREAL1:23;
     suppose A39: p2 = p01;
   then LSeg(p01,p2) /\ L3 = {p01} /\ L3 by TOPREAL1:7;
     then LSeg(p01,p2) /\ L3 = {} by Lm1,Lm7;
     hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A31,A39,TOPREAL1:21,XBOOLE_1:4
        .= {p1,p2} by ENUMSET1:41;
     suppose A40: p2 <> p01 & p2 <> p00;
A41:     L2 /\ LSeg(p00,p2) c= L2 /\ L1 by A18,XBOOLE_1:26;
        now assume p01 in L2 /\ LSeg(p00,p2);
       then p01 in LSeg(p00,p2) & p00`2 <= p2`2 by A16,EUCLID:56,XBOOLE_0:def 3
;
       then p01`2 <= p2`2 by TOPREAL1:10;
       then p2`2 = 1 by A16,Lm4,AXIOMS:21;
       hence contradiction by A16,A40,EUCLID:57;
      end;
     then {p01} <> L2 /\ LSeg(p00,p2) by ZFMISC_1:37;
then A42:  L2 /\ LSeg(p00,p2) = {} by A41,TOPREAL1:21,ZFMISC_1:39;
 A43:  LSeg(p01,p2) /\ L3 c= {p00} by A17,TOPREAL1:23,XBOOLE_1:26;
        now assume p00 in LSeg(p01,p2) /\ L3;
       then p00 in LSeg(p2,p01) & p2`2 <= p01`2 by A16,EUCLID:56,XBOOLE_0:def 3
;
       then 0 = p2`2 by A16,Lm4,TOPREAL1:10;
       hence contradiction by A16,A40,EUCLID:57;
      end;
     then {p00} <> LSeg(p01,p2) /\ L3 by ZFMISC_1:37;
     then LSeg(p01,p2) /\ L3 = {} by A43,ZFMISC_1:39;
     hence P1 /\ P2 = {p1,p2} by A31,A42,ENUMSET1:41;
    end;
    hence P1 /\ P2 = {p1,p2};
    suppose A44: p2 in L2;
     then A45: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 1 by TOPREAL1:19;
    take P1 = LSeg(p1,p11) \/ LSeg(p11,p2),P2
 = LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2));
 A46: LSeg(p11,p2) c= L2 by A44,Lm19,TOPREAL1:12;
 A47: LSeg(p2,p01) c= L2 by A44,Lm16,TOPREAL1:12;
        p11 in LSeg(p1,p11) & p11 in LSeg(p11,p2) by TOPREAL1:6;
      then LSeg(p1,p11) /\ LSeg(p11,p2) c= L4 /\ L2 &
      p11 in LSeg(p1,p11) /\ LSeg(p11,p2) & L2 /\ L4 = {p11}
       by A7,A46,TOPREAL1:24,XBOOLE_0:def 3,XBOOLE_1:27;
      then LSeg(p1,p11) /\ LSeg(p11,p2) = {p11} & (p1 <> p11 or p11 <> p2)
       by A1,ZFMISC_1:39;
    hence P1 is_an_arc_of p1,p2 by TOPREAL1:18;
    L3 is_an_arc_of p10,p00 & L1 is_an_arc_of p00,p01 by Lm4,TOPREAL1:15;
     then A48: L3 \/ L1 is_an_arc_of p10,p01 by TOPREAL1:5,23;
       L3 /\ LSeg(p01,p2) c= L3 /\ L2 by A47,XBOOLE_1:26;
 then A49: L3 /\ LSeg(p01,p2) = {} by Lm2,XBOOLE_1:3;
       p01 in LSeg(p01,p2) by TOPREAL1:6;
     then A50: L1 /\ LSeg(p01,p2) c= {p01} & L1 /\ LSeg(p01,p2) <> {}
      by A47,Lm15,TOPREAL1:21,XBOOLE_0:def 3,XBOOLE_1:27;
       (L3 \/ L1) /\ LSeg(p01,p2)
      = (L3 /\ LSeg(p01,p2)) \/ (L1 /\ LSeg(p01,p2)) by XBOOLE_1:23
     .= {p01} by A49,A50,ZFMISC_1:39;
 then A51: L3 \/ L1 \/ LSeg(p01,p2) is_an_arc_of p10,p2 by A48,TOPREAL1:16;
A52:    LSeg(p1,p10) /\ LSeg(p01,p2) c= L4 /\ L2 by A8,A47,XBOOLE_1:27;
       now assume p11 in LSeg(p1,p10) /\ LSeg(p01,p2);
      then p11 in LSeg(p10,p1) & p11 in LSeg(p01,p2) & p01`1 <= p2`1 &
       p10`2 <= p1`2 by A5,A6,A45,EUCLID:56,XBOOLE_0:def 3;
      then p11`2 <= p1`2 & p11`1 <= p2`1 by TOPREAL1:9,10;
     then A53: 1 = p1`2 & 1 = p2`2 & 1 = p1`1 & 1 = p2`1 by A5,A6,A45,Lm4,
AXIOMS:21;
      then p1 = p11 by EUCLID:57 .= p2 by A53,EUCLID:57;
      hence contradiction by A1;
     end;
     then {p11} <> LSeg(p1,p10) /\ LSeg(p01,p2) by ZFMISC_1:37;
 then A54: LSeg(p1,p10) /\ LSeg(p01,p2) = {} by A52,TOPREAL1:24,ZFMISC_1:39;
       LSeg(p1,p10) /\ (L3 \/ L1 \/ LSeg(p01,p2))
       = (LSeg(p1,p10) /\ (L3 \/ L1)) \/ (LSeg(p1,p10) /\
 LSeg(p01,p2)) by XBOOLE_1:23
      .= (LSeg(p1,p10) /\ L3) \/ (LSeg(p10,p1) /\ L1) by A54,XBOOLE_1:23
      .= {p10} by A9,A10,A13,TOPREAL1:22,ZFMISC_1:39;
    hence P2 is_an_arc_of p1,p2 by A51,TOPREAL1:17;
A55: LSeg(p1,p11) \/ LSeg(p1,p10) = L4 by A3,TOPREAL1:11;
A56: LSeg(p01,p2) \/ LSeg(p11,p2) = L2 by A44,TOPREAL1:11;
    thus P1 \/ P2
      = LSeg(p11,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p10) \/
         (L3 \/ L1 \/ LSeg(p01,p2)))) by XBOOLE_1:4
     .= LSeg(p11,p2) \/ (L4 \/ (L3 \/ L1 \/ LSeg(p01,p2))) by A55,XBOOLE_1:4
     .= LSeg(p11,p2) \/ (L4 \/ (L3 \/ L1) \/ LSeg(p01,p2)) by XBOOLE_1:4
     .= LSeg(p11,p2) \/ (L3 \/ L4 \/ L1 \/ LSeg(p01,p2)) by XBOOLE_1:4
     .= LSeg(p11,p2) \/ (L3 \/ L4 \/ (L1 \/ LSeg(p01,p2))) by XBOOLE_1:4
     .= (L1 \/ LSeg(p01,p2) \/ LSeg(p11,p2)) \/ (L3 \/ L4) by XBOOLE_1:4
     .= R^2-unit_square by A56,TOPREAL1:20,XBOOLE_1:4;
A57: {p1} = LSeg(p1,p11) /\ LSeg(p1,p10) by A3,TOPREAL1:14;
      LSeg(p1,p11) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
     by A7,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
then A58: LSeg(p1,p11) /\ L1 = {} by XBOOLE_1:3;
A59: {p2} = LSeg(p11,p2) /\ LSeg(p01,p2) by A44,TOPREAL1:14;
      LSeg(p11,p2) /\ L3 c= L2 /\ L3 & L3 /\ L2 = {} by A46,TOPREAL1:25,
XBOOLE_0:def 7,XBOOLE_1:26;
    then A60: LSeg(p11,p2) /\ L3 = {} by XBOOLE_1:3;
A61:  P1 /\ P2 = (LSeg(p1,p11) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2))))
         \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2))))
         by XBOOLE_1:23
     .= (LSeg(p1,p11) /\ LSeg(p1,p10))
        \/ (LSeg(p1,p11) /\ (L3 \/ L1 \/ LSeg(p01,p2)))
        \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ (L3 \/ L1)) \/ (LSeg(p1,p11) /\
 LSeg(p01,p2)))
        \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2))))
         by A57,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ L1)
        \/ (LSeg(p1,p11) /\ LSeg(p01,p2)))
        \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2))))
         by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2)))
        \/ ((LSeg(p11,p2) /\ LSeg(p1,p10))
        \/ (LSeg(p11,p2) /\ (L3 \/ L1 \/ LSeg(p01,p2)))) by A58,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2)))
        \/ ((LSeg(p11,p2) /\ (LSeg(p1,p10))
        \/ ((LSeg(p11,p2) /\ (L3 \/ L1)) \/ {p2}))) by A59,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2)))
        \/ ((LSeg(p11,p2) /\ (LSeg(p1,p10))
        \/ (((LSeg(p11,p2) /\ L3) \/ (LSeg(p11,p2) /\ L1)) \/
 {p2}))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2)))
        \/ ((LSeg(p11,p2) /\ LSeg(p1,p10))
        \/ ((LSeg(p11,p2) /\ L1) \/ {p2})) by A60;
A62:  now per cases;
     suppose A63: p1 = p10;
 then A64:  LSeg(p11,p2) /\ LSeg(p1,p10) = LSeg(p11,p2) /\ {p10} by TOPREAL1:7;
       p10 in LSeg(p11,p2) implies
      contradiction by A46,Lm4,TOPREAL1:10;
then A65:  LSeg(p11,p2) /\ LSeg(p1,p10) = {} by A64,Lm1;
     thus P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2))
        \/ ((LSeg(p11,p2) /\ LSeg(p1,p10))
        \/ ((LSeg(p11,p2) /\ L1) \/ {p2})) by A61,A63,TOPREAL1:22,XBOOLE_1:4
      .= {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ ((LSeg(p11,p2) /\ L1) \/
 {p2})
        by A65;
     suppose A66: p1 = p11;
   then LSeg(p1,p11) /\ L3 = {p11} /\ L3 by TOPREAL1:7;
then A67:  LSeg(p1,p11) /\ L3 = {} by Lm1,Lm12;
       p11 in LSeg(p11,p2) by TOPREAL1:6;
 then A68:  LSeg(p11,p2) /\ LSeg(p1,p10) <> {} by A66,Lm20,XBOOLE_0:def 3;
       LSeg(p11,p2) /\ LSeg(p1,p10) c= {p1} by A46,A66,TOPREAL1:24,XBOOLE_1:27;
then LSeg(p11,p2) /\ LSeg(p1,p10) = {p1} by A68,ZFMISC_1:39;
     hence P1 /\ P2 = {p1} \/ ({p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2))) \/
         ((LSeg(p11,p2) /\ L1) \/ {p2}) by A61,A67,XBOOLE_1:4
      .= {p1} \/ {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/
         ((LSeg(p11,p2) /\ L1) \/ {p2}) by XBOOLE_1:4
      .= {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ ((LSeg(p11,p2) /\ L1) \/
 {p2});
     suppose A69: p1 <> p11 & p1 <> p10;
A70:     LSeg(p1,p11) /\ L3 c= L4 /\ L3 by A7,XBOOLE_1:26;
        now assume p10 in LSeg(p1,p11) /\ L3;
       then p10 in LSeg(p1,p11) & p1`2 <= p11`2 by A5,A6,EUCLID:56,XBOOLE_0:def
3;
       then p1`2 = 0 by A5,A6,Lm4,TOPREAL1:10;
       hence contradiction by A5,A6,A69,EUCLID:57;
      end;
     then {p10} <> LSeg(p1,p11) /\ L3 by ZFMISC_1:37;
then A71:  LSeg(p1,p11) /\ L3 = {} by A70,TOPREAL1:22,ZFMISC_1:39;
 A72:  LSeg(p11,p2) /\ LSeg(p1,p10) c= {p11} by A8,A46,TOPREAL1:24,XBOOLE_1:27;
        now assume p11 in LSeg(p11,p2) /\ LSeg(p1,p10);
       then p11 in LSeg(p10,p1) & p10`2 <= p1`2 by A5,A6,EUCLID:56,XBOOLE_0:def
3;
       then p11`2 <= p1`2 by TOPREAL1:10;
       then 1 = p1`2 by A5,A6,Lm4,AXIOMS:21;
       hence contradiction by A5,A6,A69,EUCLID:57;
      end;
     then {p11} <> LSeg(p11,p2) /\ LSeg(p1,p10) by ZFMISC_1:37;
then LSeg(p11,p2) /\ LSeg(p1,p10) = {} by A72,ZFMISC_1:39;
     hence P1 /\ P2 =
 {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ ((LSeg(p11,p2) /\ L1) \/ {p2})
        by A61,A71;
    end;
      now per cases;
     suppose A73: p2 = p01;
 then A74:  LSeg(p1,p11) /\ LSeg(p01,p2) = LSeg(p1,p11) /\ {p01} by TOPREAL1:7;
       not p01 in LSeg(p1,p11) by A7,Lm4,TOPREAL1:9;
      then LSeg(p1,p11) /\ LSeg(p01,p2) = {} by A74,Lm1;
     hence P1 /\ P2 = {p1,p2} by A62,A73,ENUMSET1:41,TOPREAL1:21;
     suppose A75: p2 = p11;
A76:     LSeg(p1,p11) /\ LSeg(p01,p2) c= L4 /\ L2 by A7,A47,XBOOLE_1:27;
       p11 in LSeg(p1,p11) by TOPREAL1:6;
     then LSeg(p1,p11) /\ LSeg(p01,p2) <> {} by A75,Lm19,XBOOLE_0:def 3;
then A77:  LSeg(p1,p11) /\ LSeg(p01,p2) = {p2} by A75,A76,TOPREAL1:24,ZFMISC_1:
39;
     LSeg(p11,p2) /\ L1 = {p11} /\ L1 by A75,TOPREAL1:7;
  then LSeg(p11,p2) /\ L1 = {} by Lm1,Lm11;
     hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A62,A77,XBOOLE_1:4
       .= {p1,p2} by ENUMSET1:41;
     suppose A78: p2 <> p11 & p2 <> p01;
A79:     LSeg(p1,p11) /\ LSeg(p01,p2) c= L4 /\ L2 by A7,A47,XBOOLE_1:27;
        now assume p11 in LSeg(p1,p11) /\ LSeg(p01,p2);
       then p11 in LSeg(p01,p2) & p01`1 <= p2`1 by A45,EUCLID:56,XBOOLE_0:def 3
;
       then p11`1 <= p2`1 by TOPREAL1:9;
       then 1 = p2`1 by A45,Lm4,AXIOMS:21;
       hence contradiction by A45,A78,EUCLID:57;
      end;
     then {p11} <> LSeg(p1,p11) /\ LSeg(p01,p2) by ZFMISC_1:37;
then A80:  LSeg(p1,p11) /\ LSeg(p01,p2) = {} by A79,TOPREAL1:24,ZFMISC_1:39;
A81:
     LSeg(p11,p2) /\ L1 c= L2 /\ L1 by A46,XBOOLE_1:26;
        now assume p01 in LSeg(p11,p2) /\ L1;
       then p01 in LSeg(p2,p11) & p2`1 <= p11`1 by A45,EUCLID:56,XBOOLE_0:def 3
;
       then p2`1 = 0 by A45,Lm4,TOPREAL1:9;
       hence contradiction by A45,A78,EUCLID:57;
      end;
     then {p01} <> LSeg(p11,p2) /\ L1 by ZFMISC_1:37;
  then LSeg(p11,p2) /\ L1 = {} by A81,TOPREAL1:21,ZFMISC_1:39;
     hence P1 /\ P2 = {p1,p2} by A62,A80,ENUMSET1:41;
    end;
    hence P1 /\ P2 = {p1,p2};
    suppose A82: p2 in L3;
     then A83: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 0 by TOPREAL1:19;
    take P1 = LSeg(p1,p10) \/ LSeg(p10,p2),P2
 = LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00,p2));
 A84: LSeg(p2,p10) c= L3 by A82,Lm17,TOPREAL1:12;
 A85: LSeg(p2,p00) c= L3 by A82,Lm14,TOPREAL1:12;
       p10 in LSeg(p1,p10) & p10 in LSeg(p10,p2) by TOPREAL1:6;
     then LSeg(p1,p10) /\ LSeg(p10,p2) c= L4 /\ L3 &
      L3 /\ L4 = {p10} & LSeg(p1,p10) /\ LSeg(p10,p2) <> {}
       by A8,A84,TOPREAL1:22,XBOOLE_0:def 3,XBOOLE_1:27;
     then LSeg(p1,p10) /\ LSeg(p10,p2) = {p10} & (p1 <> p10 or p2 <> p10)
      by A1,ZFMISC_1:39;
    hence P1 is_an_arc_of p1,p2 by TOPREAL1:18;
     L2 is_an_arc_of p11,p01 & L1 is_an_arc_of p01,p00 by Lm4,TOPREAL1:15;
      then A86:  L2 \/ L1 is_an_arc_of p11,p00 by TOPREAL1:5,21;
        L2 /\ LSeg(p00,p2) c= L2 /\ L3 by A85,XBOOLE_1:26;
 then A87:  L2 /\ LSeg(p00,p2) = {} by Lm2,XBOOLE_1:3;
       p00 in LSeg(p00,p2) by TOPREAL1:6;
     then A88: L1 /\ LSeg(p00,p2) c= {p00} & L1 /\ LSeg(p00,p2) <> {}
      by A85,Lm13,TOPREAL1:23,XBOOLE_0:def 3,XBOOLE_1:27;
       (L2 \/ L1) /\ LSeg(p00,p2)
       = (L2 /\ LSeg(p00,p2)) \/ (L1 /\ LSeg(p00,p2)) by XBOOLE_1:23
      .= {p00} by A87,A88,ZFMISC_1:39;
 then A89:  L2 \/ L1 \/ LSeg(p00,p2) is_an_arc_of p11,p2 by A86,TOPREAL1:16;
A90:     LSeg(p1,p11) /\ LSeg(p00,p2) c= L4 /\ L3 by A7,A85,XBOOLE_1:27;
       now assume p10 in LSeg(p1,p11) /\ LSeg(p00,p2);
      then p10 in LSeg(p1,p11) & p10 in LSeg(p00,p2) & p1`2 <= p11`2 &
       p00`1 <= p2`1 by A5,A6,A83,EUCLID:56,XBOOLE_0:def 3;
      then p10`1 <= p2`1 & p1`2 <= p10`2 by TOPREAL1:9,10;
      then 0 = p1`2 & 0 = p2`2 & 1 = p1`1 & 1 = p2`1
       by A5,A6,A83,Lm4,AXIOMS:21;
      then p1 = p10 & p2 = p10 by EUCLID:57;
      hence contradiction by A1;
     end;
     then {p10} <> LSeg(p1,p11) /\ LSeg(p00,p2) by ZFMISC_1:37;
 then A91: LSeg(p1,p11) /\ LSeg(p00,p2) = {} by A90,TOPREAL1:22,ZFMISC_1:39;
       LSeg(p1,p11) /\ (L2 \/ L1 \/ LSeg(p00,p2))
       = (LSeg(p1,p11) /\ (L2 \/ L1)) \/ (LSeg(p1,p11) /\
 LSeg(p00,p2)) by XBOOLE_1:23
      .= (LSeg(p1,p11) /\ L2) \/ (LSeg(p1,p11) /\ L1) by A91,XBOOLE_1:23
      .= {p11} by A11,A12,A14,TOPREAL1:24,ZFMISC_1:39;
    hence P2 is_an_arc_of p1,p2 by A89,TOPREAL1:17;
A92: LSeg(p1,p10) \/ LSeg(p1,p11) = L4 by A3,TOPREAL1:11;
A93: LSeg(p00,p2) \/ LSeg(p10,p2) = L3 by A82,TOPREAL1:11;
    thus P1 \/ P2
      = LSeg(p10,p2) \/ (LSeg(p1,p10) \/ (LSeg(p1,p11) \/
         (L1 \/ L2 \/ LSeg(p00,p2)))) by XBOOLE_1:4
     .= LSeg(p10,p2) \/ (L4 \/ (L1 \/ L2 \/ LSeg(p00,p2))) by A92,XBOOLE_1:4
     .= (L1 \/ L2) \/ L4 \/ LSeg(p00,p2) \/ LSeg(p10,p2) by XBOOLE_1:4
     .= (L1 \/ L2) \/ L4 \/ L3 by A93,XBOOLE_1:4
     .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
A94: LSeg(p1,p10) /\ LSeg(p1,p11) = {p1} by A3,TOPREAL1:14;
A95: LSeg(p10,p2) /\ LSeg(p00,p2) = {p2} by A82,TOPREAL1:14;
      LSeg(p10,p2) /\ L2 c= L3 /\ L2 & L3 /\ L2 = {} by A84,TOPREAL1:25,
XBOOLE_0:def 7,XBOOLE_1:26
;
then A96: LSeg(p10,p2) /\ L2 = {} by XBOOLE_1:3;
A97:  P1 /\ P2
      = (LSeg(p1,p10) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00,p2))))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/
 LSeg(p00,p2)))) by XBOOLE_1:23
     .= (LSeg(p1,p10) /\ LSeg(p1,p11))
      \/ (LSeg(p1,p10) /\ (L2 \/ L1 \/ LSeg(p00,p2)))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/
 LSeg(p00,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ (L2 \/ L1))
      \/ (LSeg(p1,p10) /\ LSeg(p00,p2)))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00,p2))))
       by A94,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p10,p1) /\ L1)
      \/ (LSeg(p1,p10) /\ LSeg(p00,p2)))
      \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/
 LSeg(p00,p2)))) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p11))
      \/ (LSeg(p10,p2) /\ (L2 \/ L1 \/ LSeg(p00,p2)))) by A13,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p11))
      \/ ((LSeg(p10,p2) /\ (L2 \/ L1)) \/ {p2})) by A95,XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p11))
      \/ ((LSeg(p10,p2) /\ L2) \/ (LSeg(p10,p2) /\ L1) \/ {p2})) by XBOOLE_1:23
     .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2)))
      \/ ((LSeg(p10,p2) /\ LSeg(p1,p11))
      \/ ((LSeg(p10,p2) /\ L1) \/ {p2})) by A96;
A98:  now per cases;
     suppose A99: p1 = p10;
 then LSeg(p1,p10) /\ L2 = {p10} /\ L2 by TOPREAL1:7;
then A100:  LSeg(p1,p10) /\ L2 = {} by Lm1,Lm10;
 A101:  LSeg(p10,p2) /\ LSeg(p1,p11) c= {p1} by A84,A99,TOPREAL1:22,XBOOLE_1:27
;
       p10 in LSeg(p10,p2) by TOPREAL1:6;
     then LSeg(p10,p2) /\ LSeg(p1,p11) <> {} by A99,Lm18,XBOOLE_0:def 3;
  then LSeg(p10,p2) /\ LSeg(p1,p11) = {p1} by A101,ZFMISC_1:39;
     hence P1 /\ P2
       = {p1} \/ ({p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)))
        \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A97,A100,XBOOLE_1:4
      .= {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2))
        \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by XBOOLE_1:4
      .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/
 {p2});
     suppose A102: p1 = p11;
 then A103:  LSeg(p10,p2) /\ LSeg(p1,p11) = LSeg(p10,p2) /\
 {p11} by TOPREAL1:7;
       not p11 in LSeg(p10,p2) by A84,Lm4,TOPREAL1:10;
     then LSeg(p10,p2) /\ LSeg(p1,p11) = {} by A103,Lm1;
     hence P1 /\ P2
       = {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2))
        \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A97,A102,TOPREAL1:24,XBOOLE_1:4
      .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/
 {p2});
     suppose A104: p1 <> p11 & p1 <> p10;
A105:     LSeg(p1,p10) /\ L2 c= L4 /\ L2 by A8,XBOOLE_1:26;
        now assume p11 in LSeg(p1,p10) /\ L2;
       then p11 in LSeg(p10,p1) & p10`2 <= p1`2 by A5,A6,EUCLID:56,XBOOLE_0:def
3;
       then p11`2 <= p1`2 by TOPREAL1:10;
       then p1`2 = 1 by A5,A6,Lm4,AXIOMS:21;
       hence contradiction by A5,A6,A104,EUCLID:57;
      end;
     then LSeg(p1,p10) /\ L2 <> {p11} by ZFMISC_1:37;
then A106:  LSeg(p1,p10) /\ L2 = {} by A105,TOPREAL1:24,ZFMISC_1:39;
 A107:  LSeg(p10,p2) /\ LSeg(p1,p11) c= {p10} by A7,A84,TOPREAL1:22,XBOOLE_1:27
;
        now assume p10 in LSeg(p10,p2) /\ LSeg(p1,p11);
       then p10 in LSeg(p1,p11) & p1`2 <= p11`2 by A5,A6,EUCLID:56,XBOOLE_0:def
3;
       then p1`2 = 0 by A5,A6,Lm4,TOPREAL1:10;
       hence contradiction by A5,A6,A104,EUCLID:57;
      end;
     then {p10} <> LSeg(p10,p2) /\ LSeg(p1,p11) by ZFMISC_1:37;
  then LSeg(p10,p2) /\ LSeg(p1,p11) = {} by A107,ZFMISC_1:39;
     hence P1 /\ P2
       = {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/
 {p2})
 by A97,A106;
    end;
      now per cases;
     suppose A108: p2 = p00;
 then A109:  LSeg(p1,p10) /\ LSeg(p00,p2) = LSeg(p1,p10) /\ {p00} by TOPREAL1:7
;
       not p00 in LSeg(p1,p10) by A8,Lm4,TOPREAL1:9;
     then LSeg(p1,p10) /\ LSeg(p00,p2) = {} by A109,Lm1;
     hence P1 /\ P2 = {p1,p2} by A98,A108,ENUMSET1:41,TOPREAL1:23;
     suppose A110: p2 = p10;
A111:     LSeg(p1,p10) /\ LSeg(p00,p2) c= L4 /\ L3 by A8,A85,XBOOLE_1:27;
       p10 in LSeg(p1,p10) by TOPREAL1:6;
     then LSeg(p1,p10) /\ LSeg(p00,p2) <> {} by A110,Lm17,XBOOLE_0:def 3;
then A112:  LSeg(p1,p10) /\
 LSeg(p00,p2) = {p2} by A110,A111,TOPREAL1:22,ZFMISC_1:39;
     LSeg(p10,p2) /\ L1 = {p10} /\ L1 by A110,TOPREAL1:7;
  then LSeg(p10,p2) /\ L1 = {} by Lm1,Lm9;
     hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A98,A112,XBOOLE_1:4
      .= {p1,p2} by ENUMSET1:41;
     suppose A113: p2 <> p10 & p2 <> p00;
A114:     LSeg(p1,p10) /\ LSeg(p00,p2) c= L4 /\ L3 by A8,A85,XBOOLE_1:27;
        now assume p10 in LSeg(p1,p10) /\ LSeg(p00,p2);
       then p10 in LSeg(p00,p2) & p00`1 <= p2`1 by A83,EUCLID:56,XBOOLE_0:def 3
;
       then p10`1 <= p2`1 by TOPREAL1:9;
       then p2`1 = 1 by A83,Lm4,AXIOMS:21;
       hence contradiction by A83,A113,EUCLID:57;
      end;
     then {p10} <> LSeg(p1,p10) /\ LSeg(p00,p2) by ZFMISC_1:37;
then A115:  LSeg(p1,p10) /\ LSeg(p00,p2) = {} by A114,TOPREAL1:22,ZFMISC_1:39;
A116:
     LSeg(p10,p2) /\ L1 c= L3 /\ L1 by A84,XBOOLE_1:26;
        now assume p00 in LSeg(p10,p2) /\ L1;
       then p00 in LSeg(p2,p10) & p2`1 <= p10`1 by A83,EUCLID:56,XBOOLE_0:def 3
;
       then p2`1 = 0 by A83,Lm4,TOPREAL1:9;
       hence contradiction by A83,A113,EUCLID:57;
      end;
     then {p00} <> LSeg(p10,p2) /\ L1 by ZFMISC_1:37;
  then LSeg(p10,p2) /\ L1 = {} by A116,TOPREAL1:23,ZFMISC_1:39;
     hence P1 /\ P2 = {p1,p2} by A98,A115,ENUMSET1:41;
    end;
    hence P1 /\ P2 = {p1,p2};
    suppose A117: p2 in L4;
     then consider q such that
A118: q = p2 and A119: q`1 = 1 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:19;
A120:     p = |[p`1,p`2]| & q = |[q`1,q`2]| by EUCLID:57;
 A121: LSeg(p11,p2) c= L4 by A117,Lm20,TOPREAL1:12;
 A122: LSeg(p10,p2) c= L4 by A117,Lm18,TOPREAL1:12;
 A123: LSeg(p1,p2) c= L4 by A3,A117,TOPREAL1:12;
       now per cases by A1,A5,A6,A118,A119,A120,REAL_1:def 5;
      suppose A124: p`2 < q`2;
       take P1 = LSeg(p1,p2),P2 = LSeg(p1,p10) \/ (L3 \/ L1 \/ L2 \/
 LSeg(p11,p2));
       thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
 A125:    now assume
A126:      LSeg(p1,p10) /\ LSeg(p11,p2) <> {};
         consider a being Element of LSeg(p1,p10) /\ LSeg(p11,p2);
         reconsider p = a as Point of TOP-REAL 2 by A126,TARSKI:def 3;
           p in LSeg(p10,p1) & p in LSeg(p2,p11) & p10`2 <= p1`2 & p2`2 <= p11
`2
          by A5,A6,A118,A119,A126,EUCLID:56,XBOOLE_0:def 3;
         then p`2 <= p1`2 & p2`2 <= p`2 by TOPREAL1:10;
         hence contradiction by A5,A118,A124,AXIOMS:22;
        end;
A127:       LSeg(p1,p10) /\ L2 c= L4 /\ L2 by A8,XBOOLE_1:26;
          now assume p11 in LSeg(p1,p10) /\ L2;
         then p11 in LSeg(p10,p1) & p10`2 <=
 p1`2 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         then p11`2 <= p1`2 by TOPREAL1:10;
         hence contradiction by A5,A6,A119,A124,Lm4,AXIOMS:21;
        end;
       then {p11} <> LSeg(p1,p10) /\ L2 by ZFMISC_1:37;
 then A128:   LSeg(p1,p10) /\ L2 = {} by A127,TOPREAL1:24,ZFMISC_1:39;
A129:
       LSeg(p1,p10) /\ L3 c= L4 /\ L3 by A8,XBOOLE_1:26;
         p10 in LSeg(p1,p10) by TOPREAL1:6;
       then A130: LSeg(p1,p10) /\ L3 <> {} by Lm17,XBOOLE_0:def 3;
         LSeg(p1,p10) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
        by A8,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
 then A131:   LSeg(p1,p10) /\ L1 = {} by XBOOLE_1:3;
 A132:   LSeg(p1,p10) /\ (L3 \/ L1 \/ L2 \/ LSeg(p11,p2))
         = LSeg(p1,p10) /\ (L3 \/ L1 \/ L2) \/ (LSeg(p1,p10) /\ LSeg(p11,p2))
            by XBOOLE_1:23
        .= LSeg(p1,p10) /\ (L3 \/ L1) \/ (LSeg(p1,p10) /\ L2) by A125,XBOOLE_1:
23
        .= LSeg(p1,p10) /\ L3 \/ (LSeg(p1,p10) /\ L1) by A128,XBOOLE_1:23
        .= {p10} by A129,A130,A131,TOPREAL1:22,ZFMISC_1:39;
      L3 is_an_arc_of p10,p00 by Lm4,TOPREAL1:15;
       then A133:   L3 \/ L1 is_an_arc_of p10,p01 by TOPREAL1:16,23;
      (L3 \/ L1) /\ L2 = L3 /\ L2 \/ L1 /\ L2 by XBOOLE_1:23
                     .= {p01} by Lm2,TOPREAL1:21;
 then A134:   L3 \/ L1 \/ L2 is_an_arc_of p10,p11 by A133,TOPREAL1:16;
  A135:   L2 /\ LSeg(p11,p2) c= {p11} by A121,TOPREAL1:24,XBOOLE_1:26;
         p11 in LSeg(p11,p2) by TOPREAL1:6;
       then L2 /\ LSeg(p11,p2) <> {} by Lm19,XBOOLE_0:def 3;
 then A136:   L2 /\ LSeg(p11,p2) = {p11} by A135,ZFMISC_1:39;
  A137:   L3 /\ LSeg(p11,p2) c= {p10} by A121,TOPREAL1:22,XBOOLE_1:26;
          now assume p10 in L3 /\ LSeg(p11,p2);
         then p10 in LSeg(p2,p11) & p2`2 <= p11`2 by A118,A119,EUCLID:56,
XBOOLE_0:def 3;
         hence contradiction by A6,A118,A124,Lm4,TOPREAL1:10;
        end;
       then {p10} <> L3 /\ LSeg(p11,p2) by ZFMISC_1:37;
 then A138:   L3 /\ LSeg(p11,p2) = {} by A137,ZFMISC_1:39;
         L1 /\ LSeg(p11,p2) c= L1 /\ L4 & L1 /\ L4 = {}
        by A121,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
 then A139:   L1 /\ LSeg(p11,p2) = {} by XBOOLE_1:3;
         (L3 \/ L1 \/ L2) /\ LSeg(p11,p2)
         = (L3 \/ L1) /\ LSeg(p11,p2) \/ L2 /\ LSeg(p11,p2) by XBOOLE_1:23
        .= (L3 /\ LSeg(p11,p2)) \/ (L1 /\ LSeg(p11,p2)) \/
 {p11} by A136,XBOOLE_1:23
        .= {p11} by A138,A139;
       then L3 \/ L1 \/ L2 \/
 LSeg(p11,p2) is_an_arc_of p10,p2 by A134,TOPREAL1:16;
       hence P2 is_an_arc_of p1,p2 by A132,TOPREAL1:17;
       thus P1 \/ P2
         = (L3 \/ L1 \/ L2) \/ LSeg(p11,p2) \/ (LSeg(p1,p10) \/
 LSeg(p1,p2)) by XBOOLE_1:4
        .= (L3 \/ L1 \/ L2) \/ (LSeg(p10,p1) \/ LSeg(p1,p2) \/
 LSeg(p2,p11)) by XBOOLE_1:4
        .= (L3 \/ L1 \/ L2) \/ L4 by A3,A117,TOPREAL1:13
        .= (L3 \/ (L1 \/ L2)) \/ L4 by XBOOLE_1:4
        .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
         p1 in LSeg(p1,p2) & p1 in LSeg(p1,p10) by TOPREAL1:6;
       then p1 in LSeg(p1,p2) /\ LSeg(p1,p10) by XBOOLE_0:def 3;
  then A140:   {p1} c= LSeg(p1,p2) /\ LSeg(p1,p10) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p1,p10) c= {p1}
        proof
         let a; assume A141: a in LSeg(p1,p2) /\ LSeg(p1,p10);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p1,p2) & p in LSeg(p10,p1) & p1`1 <= p2`1 & p1`2 <= p2`2
          & p10`2 <= p1`2 by A5,A6,A118,A119,A124,A141,EUCLID:56,XBOOLE_0:def 3
;
         then p`1 <= p2`1 & p1`1 <= p`1 & p`2 <= p1`2 & p1`2 <= p`2
          by TOPREAL1:9,10;
         then p1`2 = p`2 & p`1 = 1 by A5,A6,A118,A119,AXIOMS:21;
         then p = |[ 1, p1`2]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
         hence a in {p1} by TARSKI:def 1;
        end;
 then A142:   LSeg(p1,p2) /\ LSeg(p1,p10) = {p1} by A140,XBOOLE_0:def 10;
         p2 in LSeg(p1,p2) & p2 in LSeg(p11,p2) by TOPREAL1:6;
       then p2 in LSeg(p1,p2) /\ LSeg(p11,p2) by XBOOLE_0:def 3;
  then A143:   {p2} c= LSeg(p1,p2) /\ LSeg(p11,p2) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p11,p2) c= {p2}
        proof
         let a; assume A144: a in LSeg(p1,p2) /\ LSeg(p11,p2);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p1,p2) & p in LSeg(p2,p11) & p1`1 <= p2`1 & p1`2 <= p2`2
          & p2`2 <= p11`2 by A5,A6,A118,A119,A124,A144,EUCLID:56,XBOOLE_0:def 3
;
         then p`1 <= p2`1 & p1`1 <= p`1 & p`2 <= p2`2 & p2`2 <= p`2
          by TOPREAL1:9,10;
         then p2`2 = p`2 & p`1 = 1 by A5,A6,A118,A119,AXIOMS:21;
         then p = |[ 1, p2`2]| by EUCLID:57 .= p2 by A118,A119,EUCLID:57;
         hence a in {p2} by TARSKI:def 1;
        end;
 then A145:   LSeg(p1,p2) /\ LSeg(p11,p2) = {p2} by A143,XBOOLE_0:def 10;
         LSeg(p1,p2) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
        by A123,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
 then A146:   LSeg(p1,p2) /\ L1 = {} by XBOOLE_1:3;
  A147:   P1 /\ P2
         = {p1} \/ LSeg(p1,p2) /\ (L3 \/ L1 \/ L2 \/
 LSeg(p11,p2)) by A142,XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L3 \/ L1 \/ L2)) \/ {p2}) by A145,XBOOLE_1
:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L3 \/ L1)) \/ (LSeg(p1,p2) /\ L2) \/ {p2})
            by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ (LSeg(p1,p2) /\ L1)
            \/ (LSeg(p1,p2) /\ L2) \/ {p2}) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/ {p2}))
            by A146,XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/ {p2})
            by XBOOLE_1:4;
A148:        LSeg(p1,p2) /\ L3 c= L4 /\ L3 by A123,XBOOLE_1:26;
A149:      now per cases;
         suppose A150: p1 = p10;
         then p10 in LSeg(p1,p2) by TOPREAL1:6;
         then LSeg(p1,p2) /\ L3 <> {} by Lm17,XBOOLE_0:def 3;
         then LSeg(p1,p2) /\ L3 = {p1} by A148,A150,TOPREAL1:22,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A147;
         suppose A151: p1 <> p10;
            now assume p10 in LSeg(p1,p2) /\ L3;
           then p10 in LSeg(p1,p2) & p1`2 <= p2`2 by A5,A118,A124,XBOOLE_0:def
3;
           then p1`2 = 0 by A5,A6,Lm4,TOPREAL1:10;
           hence contradiction by A5,A6,A151,EUCLID:57;
          end;
         then {p10} <> LSeg(p1,p2) /\ L3 by ZFMISC_1:37;
         then LSeg(p1,p2) /\ L3 = {} by A148,TOPREAL1:22,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A147;
        end;
A152:        LSeg(p1,p2) /\ L2 c= L4 /\ L2 by A123,XBOOLE_1:26;
          now per cases;
         suppose A153: p2 = p11;
         then p11 in LSeg(p1,p2) by TOPREAL1:6;
         then LSeg(p1,p2) /\ L2 <> {} by Lm19,XBOOLE_0:def 3;
         then LSeg(p1,p2) /\ L2 = {p2} by A152,A153,TOPREAL1:24,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A149,ENUMSET1:41;
         suppose A154: p2 <> p11;
            now assume p11 in LSeg(p1,p2) /\ L2;
           then p11 in LSeg(p1,p2) & p1`2 <= p2`2 by A5,A118,A124,XBOOLE_0:def
3;
           then p11`2 <= p2`2 by TOPREAL1:10;
           then p2`2 = 1 by A118,A119,Lm4,AXIOMS:21;
           hence contradiction by A118,A119,A154,EUCLID:57;
          end;
         then {p11} <> LSeg(p1,p2) /\ L2 by ZFMISC_1:37;
         then LSeg(p1,p2) /\ L2 = {} by A152,TOPREAL1:24,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A149,ENUMSET1:41;
        end;
       hence P1 /\ P2 = {p1,p2};
      suppose A155: q`2 < p`2;
       take P1 = LSeg(p1,p2),P2 = LSeg(p1,p11) \/ (L2 \/ L1 \/ L3 \/
 LSeg(p10,p2));
       thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:15;
 A156:    now assume
A157:      LSeg(p1,p11) /\ LSeg(p10,p2) <> {};
         consider a being Element of LSeg(p1,p11) /\ LSeg(p10,p2);
         reconsider p = a as Point of TOP-REAL 2 by A157,TARSKI:def 3;
           p in LSeg(p1,p11) & p in LSeg(p10,p2) & p1`2 <= p11`2 & p10`2 <= p2
`2
          by A5,A6,A118,A119,A157,EUCLID:56,XBOOLE_0:def 3;
         then p`2 <= p2`2 & p1`2 <= p`2 by TOPREAL1:10;
         hence contradiction by A5,A118,A155,AXIOMS:22;
        end;
A158:       LSeg(p1,p11) /\ L3 c= L4 /\ L3 by A7,XBOOLE_1:26;
          now assume p10 in LSeg(p1,p11) /\ L3;
         then p10 in LSeg(p1,p11) & p1`2 <=
 p11`2 by A5,A6,EUCLID:56,XBOOLE_0:def 3;
         hence contradiction by A5,A119,A155,Lm4,TOPREAL1:10;
        end;
       then {p10} <> LSeg(p1,p11) /\ L3 by ZFMISC_1:37;
 then A159:   LSeg(p1,p11) /\ L3 = {} by A158,TOPREAL1:22,ZFMISC_1:39;
A160:  LSeg(p1,p11) /\ L2 c= L4 /\ L2 by A7,XBOOLE_1:26;
         p11 in LSeg(p1,p11) by TOPREAL1:6;
       then A161: LSeg(p1,p11) /\ L2 <> {} by Lm19,XBOOLE_0:def 3;
         LSeg(p1,p11) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
        by A7,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
 then A162:   LSeg(p1,p11) /\ L1 = {} by XBOOLE_1:3;
 A163:   LSeg(p1,p11) /\ (L2 \/ L1 \/ L3 \/ LSeg(p10,p2))
         = LSeg(p1,p11) /\ (L2 \/ L1 \/ L3) \/ (LSeg(p1,p11) /\ LSeg(p10,p2))
            by XBOOLE_1:23
        .= LSeg(p1,p11) /\ (L2 \/ L1) \/ (LSeg(p1,p11) /\ L3) by A156,XBOOLE_1:
23
        .= LSeg(p1,p11) /\ L2 \/ (LSeg(p1,p11) /\ L1) by A159,XBOOLE_1:23
        .= {p11} by A160,A161,A162,TOPREAL1:24,ZFMISC_1:39;
      L2 is_an_arc_of p11,p01 by Lm4,TOPREAL1:15;
       then A164:   L2 \/ L1 is_an_arc_of p11,p00 by TOPREAL1:16,21;
         (L2 \/ L1) /\ L3 = L3 /\ L2 \/ L1 /\ L3 by XBOOLE_1:23
                     .= {p00} by Lm2,TOPREAL1:23;
 then A165:   L2 \/ L1 \/ L3 is_an_arc_of p11,p10 by A164,TOPREAL1:16;
  A166:   L3 /\ LSeg(p10,p2) c= {p10} by A122,TOPREAL1:22,XBOOLE_1:26;
         p10 in LSeg(p10,p2) by TOPREAL1:6;
       then L3 /\ LSeg(p10,p2) <> {} by Lm17,XBOOLE_0:def 3;
 then A167:   L3 /\ LSeg(p10,p2) = {p10} by A166,ZFMISC_1:39;
  A168:   L2 /\ LSeg(p10,p2) c= {p11} by A122,TOPREAL1:24,XBOOLE_1:26;
          now assume p11 in L2 /\ LSeg(p10,p2);
         then p11 in LSeg(p10,p2) & p10`2 <= p2`2 by A118,A119,EUCLID:56,
XBOOLE_0:def 3;
         then p11`2 <= p2`2 by TOPREAL1:10;
         hence contradiction by A6,A118,A119,A155,Lm4,AXIOMS:21;
        end;
       then {p11} <> L2 /\ LSeg(p10,p2) by ZFMISC_1:37;
 then A169:   L2 /\ LSeg(p10,p2) = {} by A168,ZFMISC_1:39;
         L1 /\ LSeg(p10,p2) c= L1 /\ L4 & L1 /\ L4 = {}
        by A122,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
 then A170:   L1 /\ LSeg(p10,p2) = {} by XBOOLE_1:3;
         (L2 \/ L1 \/ L3) /\ LSeg(p10,p2)
         = (L2 \/ L1) /\ LSeg(p10,p2) \/ L3 /\ LSeg(p10,p2) by XBOOLE_1:23
        .= (L2 /\ LSeg(p10,p2)) \/ (L1 /\ LSeg(p10,p2)) \/
 {p10} by A167,XBOOLE_1:23
        .= {p10} by A169,A170;
       then L2 \/ L1 \/ L3 \/
 LSeg(p10,p2) is_an_arc_of p11,p2 by A165,TOPREAL1:16
;
       hence P2 is_an_arc_of p1,p2 by A163,TOPREAL1:17;
       thus P1 \/ P2
         = (L2 \/ L1 \/ L3) \/ LSeg(p10,p2) \/ (LSeg(p1,p11) \/
 LSeg(p1,p2)) by XBOOLE_1:4
        .= (L2 \/ L1 \/ L3) \/ (LSeg(p10,p2) \/ (LSeg(p1,p2) \/
 LSeg(p1,p11))) by XBOOLE_1:4
        .= (L1 \/ L2 \/ L3) \/ L4 by A3,A117,TOPREAL1:13
        .= R^2-unit_square by TOPREAL1:20,XBOOLE_1:4;
         p1 in LSeg(p1,p2) & p1 in LSeg(p1,p11) by TOPREAL1:6;
       then p1 in LSeg(p1,p2) /\ LSeg(p1,p11) by XBOOLE_0:def 3;
  then A171:   {p1} c= LSeg(p1,p2) /\ LSeg(p1,p11) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p1,p11) c= {p1}
        proof
         let a; assume A172: a in LSeg(p1,p2) /\ LSeg(p1,p11);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p2,p1) & p in LSeg(p1,p11) & p2`1 <= p1`1 & p2`2 <= p1`2
          & p1`2 <= p11`2 by A5,A6,A118,A119,A155,A172,EUCLID:56,XBOOLE_0:def 3
;
         then p`1 <= p1`1 & p2`1 <= p`1 & p`2 <= p1`2 & p1`2 <= p`2
          by TOPREAL1:9,10;
         then p1`2 = p`2 & p`1 = 1 by A5,A6,A118,A119,AXIOMS:21;
         then p = |[ 1, p1`2]| by EUCLID:57 .= p1 by A5,A6,EUCLID:57;
         hence a in {p1} by TARSKI:def 1;
        end;
 then A173:   LSeg(p1,p2) /\ LSeg(p1,p11) = {p1} by A171,XBOOLE_0:def 10;
         p2 in LSeg(p1,p2) & p2 in LSeg(p10,p2) by TOPREAL1:6;
       then p2 in LSeg(p1,p2) /\ LSeg(p10,p2) by XBOOLE_0:def 3;
  then A174:   {p2} c= LSeg(p1,p2) /\ LSeg(p10,p2) by ZFMISC_1:37;
         LSeg(p1,p2) /\ LSeg(p10,p2) c= {p2}
        proof
         let a; assume A175: a in LSeg(p1,p2) /\ LSeg(p10,p2);
         then reconsider p = a as Point of TOP-REAL 2;
           p in LSeg(p2,p1) & p in LSeg(p10,p2) & p2`1 <= p1`1 & p2`2 <= p1`2
          & p10`2 <= p2`2 by A5,A6,A118,A119,A155,A175,EUCLID:56,XBOOLE_0:def 3
;
         then p`1 <= p1`1 & p2`1 <= p`1 & p`2 <= p2`2 & p2`2 <= p`2
          by TOPREAL1:9,10;
         then p2`2 = p`2 & p`1 = 1 by A5,A6,A118,A119,AXIOMS:21;
         then p = |[ 1, p2`2]| by EUCLID:57 .= p2 by A118,A119,EUCLID:57;
         hence a in {p2} by TARSKI:def 1;
        end;
 then A176:   LSeg(p1,p2) /\ LSeg(p10,p2) = {p2} by A174,XBOOLE_0:def 10;
         LSeg(p1,p2) /\ L1 c= L4 /\ L1 & L1 /\ L4 = {}
        by A123,TOPREAL1:26,XBOOLE_0:def 7,XBOOLE_1:26;
 then A177:   LSeg(p1,p2) /\ L1 = {} by XBOOLE_1:3;
  A178:   P1 /\ P2
         = {p1} \/ LSeg(p1,p2) /\ (L2 \/ L1 \/ L3 \/
 LSeg(p10,p2)) by A173,XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L2 \/ L1 \/ L3)) \/ {p2}) by A176,XBOOLE_1
:23
        .= {p1} \/ ((LSeg(p1,p2) /\ (L2 \/ L1)) \/ (LSeg(p1,p2) /\ L3) \/ {p2})
            by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L2) \/ (LSeg(p1,p2) /\ L1)
            \/ (LSeg(p1,p2) /\ L3) \/ {p2}) by XBOOLE_1:23
        .= {p1} \/ ((LSeg(p1,p2) /\ L2) \/ ((LSeg(p1,p2) /\ L3) \/ {p2}))
            by A177,XBOOLE_1:4
        .= {p1} \/ (LSeg(p1,p2) /\ L2) \/ ((LSeg(p1,p2) /\ L3) \/ {p2})
            by XBOOLE_1:4;
A179:        LSeg(p1,p2) /\ L2 c= L4 /\ L2 by A123,XBOOLE_1:26;
A180:      now per cases;
         suppose A181: p1 = p11;
         then p11 in LSeg(p1,p2) by TOPREAL1:6;
         then LSeg(p1,p2) /\ L2 <> {} by Lm19,XBOOLE_0:def 3;
         then LSeg(p1,p2) /\ L2 = {p1} by A179,A181,TOPREAL1:24,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L3) \/ {p2}) by A178;
         suppose A182: p1 <> p11;
            now assume p11 in LSeg(p1,p2) /\ L2;
           then p11 in LSeg(p2,p1) & p2`2 <= p1`2
            by A5,A118,A155,XBOOLE_0:def 3;
           then p11`2 <= p1`2 by TOPREAL1:10;
           then p1`2 = 1 by A5,A6,Lm4,AXIOMS:21;
           hence contradiction by A5,A6,A182,EUCLID:57;
          end;
         then {p11} <> LSeg(p1,p2) /\ L2 by ZFMISC_1:37;
         then LSeg(p1,p2) /\ L2 = {} by A179,TOPREAL1:24,ZFMISC_1:39;
         hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L3) \/ {p2}) by A178;
        end;
A183:        LSeg(p1,p2) /\ L3 c= L4 /\ L3 by A123,XBOOLE_1:26;
          now per cases;
         suppose A184: p2 = p10;
         then p10 in LSeg(p1,p2) by TOPREAL1:6;
         then LSeg(p1,p2) /\ L3 <> {} by Lm17,XBOOLE_0:def 3;
         then LSeg(p1,p2) /\ L3 = {p2} by A183,A184,TOPREAL1:22,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A180,ENUMSET1:41;
         suppose A185: p2 <> p10;
            now assume p10 in LSeg(p1,p2) /\ L3;
           then p10 in LSeg(p2,p1) & p2`2 <= p1`2
            by A5,A118,A155,XBOOLE_0:def 3;
           then p2`2 = 0 by A118,A119,Lm4,TOPREAL1:10;
           hence contradiction by A118,A119,A185,EUCLID:57;
          end;
         then {p10} <> LSeg(p1,p2) /\ L3 by ZFMISC_1:37;
         then LSeg(p1,p2) /\ L3 = {} by A183,TOPREAL1:22,ZFMISC_1:39;
         hence P1 /\ P2 = {p1,p2} by A180,ENUMSET1:41;
        end;
       hence P1 /\ P2 = {p1,p2};
     end;
    hence ex P1,P2 being non empty Subset of TOP-REAL 2
    st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
     R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2};
 end;

theorem Th1:
 p1 <> p2 & p1 in R^2-unit_square & p2 in R^2-unit_square implies
  ex P1, P2 being non empty Subset of TOP-REAL 2
   st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
   R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2}
 proof
   assume that
A1: p1 <> p2 and
A2: p1 in R^2-unit_square & p2 in R^2-unit_square;
A3: p1 in L1 \/ L2 or p1 in L3 \/ L4 by A2,TOPREAL1:20,XBOOLE_0:def 2;
  per cases by A3,XBOOLE_0:def 2;
  suppose p1 in L1;
   hence thesis by A1,A2,Lm23;
  suppose p1 in L2;
   hence thesis by A1,A2,Lm24;
  suppose p1 in L3;
   hence thesis by A1,A2,Lm25;
  suppose p1 in L4;
   hence thesis by A1,A2,Lm26;
 end;

theorem Th2:
R^2-unit_square is compact
proof
 consider P1,P2 being non empty Subset of TOP-REAL 2 such that
 A1: P1 is_S-P_arc & P2 is_S-P_arc & R^2-unit_square = P1 \/ P2 by TOPREAL1:34;
    consider f being map of I[01], (TOP-REAL 2)|P1 such that
         A2:    f is_homeomorphism by A1,TOPREAL1:36;
 A3:    f is continuous by A2,TOPS_2:def 5;
   consider f0 being map of I[01], (TOP-REAL 2)|P2 such that
         A4:    f0 is_homeomorphism by A1,TOPREAL1:36;
 A5:    f0 is continuous by A4,TOPS_2:def 5;
 A6: I[01] is compact by HEINE:11,TOPMETR:27;
 A7: rng f = [#]((TOP-REAL 2)|P1) by A2,TOPS_2:def 5;
  reconsider P1 as non empty Subset of TOP-REAL 2;
   (TOP-REAL 2)|P1 is compact by A3,A6,A7,COMPTS_1:23;
 then A8: P1 is compact by COMPTS_1:12;
 A9: rng f0 = [#]((TOP-REAL 2)|P2) by A4,TOPS_2:def 5;
  reconsider P2 as non empty Subset of TOP-REAL 2;
   (TOP-REAL 2)|P2 is compact by A5,A6,A9,COMPTS_1:23;
 then P2 is compact by COMPTS_1:12;
 hence thesis by A1,A8,COMPTS_1:19;
end;

theorem Th3:
for Q, P being non empty Subset of TOP-REAL 2
 for f being map of (TOP-REAL 2)|Q, (TOP-REAL 2)|P st f is_homeomorphism &
  Q is_an_arc_of q1,q2 holds
  for p1, p2 st p1 = f.q1 & p2 = f.q2 holds P is_an_arc_of p1,p2
proof
 let Q, P be non empty Subset of TOP-REAL 2;
 let f be map of (TOP-REAL 2)|Q, (TOP-REAL 2)|P; assume that
A1: f is_homeomorphism and
A2: Q is_an_arc_of q1,q2;
 let p1, p2 such that
A3: p1 = f.q1 & p2 = f.q2;
  reconsider f as map of (TOP-REAL 2)|Q, (TOP-REAL 2)|P;
  consider f1 being map of I[01], (TOP-REAL 2)|Q such that
A4: f1 is_homeomorphism & f1.0 = q1 & f1.1 = q2 by A2,TOPREAL1:def 2;
  set g1 = f*f1;
A5: g1 is_homeomorphism by A1,A4,TOPS_2:71;
A6: dom f1 = the carrier of I[01] & the carrier of I[01] = [.0,1.]
     by BORSUK_1:83,FUNCT_2:def 1;
then 0 in dom f1 by RCOMP_1:15;
then A7: g1.0 = p1 by A3,A4,FUNCT_1:23;
   1 in dom f1 by A6,RCOMP_1:15;
   then g1.1 = p2 by A3,A4,FUNCT_1:23;
  hence thesis by A5,A7,TOPREAL1:def 2;
end;

definition let P be Subset of TOP-REAL 2;
  attr P is being_simple_closed_curve means
   :Def1:
 ex f being map of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P
    st f is_homeomorphism;
  synonym P is_simple_closed_curve;
end;

definition
 cluster R^2-unit_square -> being_simple_closed_curve;
coherence
  proof
    set T = (TOP-REAL 2)|R^2-unit_square;
    take f = id T;
    thus dom f = [#]T by TOPS_2:51;
A1: f = id the carrier of T by GRCAT_1:def 11;
    hence
A2:   rng f = the carrier of T by RELAT_1:71
           .= [#]T by PRE_TOPC:12;
    thus
A3:   f is one-to-one by A1,FUNCT_1:52;
    thus
A4:   f is continuous
    proof
      let V be Subset of T such that
A5:       V is closed;
      thus f"V is closed by A1,A5,BORSUK_1:4;
    end;
      f" = f
    proof
      thus f" = (f qua Function)" by A2,A3,TOPS_2:def 4
        .= (id the carrier of T)" by GRCAT_1:def 11
        .= id the carrier of T by FUNCT_1:67
        .= f by GRCAT_1:def 11;
    end;
    hence thesis by A4;
  end;
end;

definition
 cluster being_simple_closed_curve Subset of TOP-REAL 2;
existence
  proof
    take R^2-unit_square;
    thus thesis;
  end;
end;

definition
 mode Simple_closed_curve is being_simple_closed_curve Subset of TOP-REAL 2;
end;

theorem Th4:
for P being non empty Subset of TOP-REAL 2 st
 P is_simple_closed_curve ex p1,p2 st p1 <> p2 & p1 in P & p2 in P
 proof
  let P be non empty Subset of TOP-REAL 2;
  assume P is_simple_closed_curve;
  then consider f being map of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P
   such that A1: f is_homeomorphism by Def1;
  A2: rng f = [#]((TOP-REAL 2)|P) by A1,TOPS_2:def 5 .= P by PRE_TOPC:def 10;
  A3: p00`1 = 0 & p00`2 = 0 & p11`1 = 1 & p11`2 = 1 by EUCLID:56;
  reconsider RS = R^2-unit_square
             as non empty Subset of TOP-REAL 2;
  reconsider f as map of (TOP-REAL 2)|RS, (TOP-REAL 2)|P;
     dom f = [#]((TOP-REAL 2)|RS) by TOPS_2:51
       .= R^2-unit_square by PRE_TOPC:def 10;
  then A4: p00 in dom f & p11 in dom f by A3,TOPREAL1:def 3;
  set p1 = f.p00, p2 = f.(p11);
    rng f = [#]((TOP-REAL 2)|P) & [#]((TOP-REAL 2)|P) c= [#]
(TOP-REAL 2) & p1 in rng f
   & p2 in rng f by A1,A4,FUNCT_1:def 5,PRE_TOPC:def 9,TOPS_2:def 5;
  then reconsider p1, p2 as Point of TOP-REAL 2 by PRE_TOPC:12;
  take p1, p2;
    p00 <> p11 & f is one-to-one by A1,A3,TOPS_2:def 5;
  hence p1 <> p2 by A4,FUNCT_1:def 8;
  thus thesis by A2,A4,FUNCT_1:def 5;
 end;

Lm27:
 for P, P1, P2 being non empty Subset of TOP-REAL 2 st
  p1 <> p2 & p1 in P & p2 in P &
    P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
    P = P1 \/ P2 & P1 /\ P2 = {p1,p2}
   holds P is_simple_closed_curve
 proof
let P, P1, P2 be non empty Subset of TOP-REAL 2 such that
   p1 <> p2 & p1 in P & p2 in P and
A1: P1 is_an_arc_of p1,p2 and
A2: P2 is_an_arc_of p1,p2 and
A3: P = P1 \/ P2 and
A4: P1 /\ P2 = {p1,p2};
  consider f1 being map of I[01], (TOP-REAL 2)|P1 such that
 A5: f1 is_homeomorphism and A6: f1.0 = p1 & f1.1 = p2 by A1,TOPREAL1:def 2;
  consider f2 being map of I[01], (TOP-REAL 2)|P2 such that
 A7: f2 is_homeomorphism and A8: f2.0 = p1 & f2.1 = p2 by A2,TOPREAL1:def 2;
      consider h1,h2 such that A9: h1 is_S-Seq & h2 is_S-Seq and
A10: R^2-unit_square = L~h1 \/ L~h2 & L~h1 /\ L~h2 = {p00, p11} and
A11: h1/.1 = p00 & h1/.len h1=p11 and
A12: h2/.1 = p00 & h2/.len h2 = p11 by TOPREAL1:30;
  reconsider L1 = L~h1, L2 = L~h2
    as non empty Subset of TOP-REAL 2 by A10;
    L1 is_an_arc_of p00,p11 by A9,A11,TOPREAL1:31;
  then consider g1 being map of I[01], (TOP-REAL 2)|L1 such that
A13: g1 is_homeomorphism & g1.0 = p00 & g1.1 = p11 by TOPREAL1:def 2;
    L2 is_an_arc_of p00,p11 by A9,A12,TOPREAL1:31;
  then consider g2 being map of I[01], (TOP-REAL 2)|L2 such that
A14: g2 is_homeomorphism & g2.0 = p00 & g2.1 = p11 by TOPREAL1:def 2;
  A15: [#]((TOP-REAL 2)|P1) = P1 & [#]((TOP-REAL 2)|P) = P & P1 c= P
   by A3,PRE_TOPC:def 10,XBOOLE_1:7;
  then the carrier of ((TOP-REAL 2)|P1) c= [#]((TOP-REAL 2)|P) by PRE_TOPC:12;
  then A16: the carrier of ((TOP-REAL 2)|P1) c= the carrier of ((TOP-REAL 2)|P)
   by PRE_TOPC:12;
  reconsider P' = P, P1' = P1, P2' = P2
   as non empty Subset of TOP-REAL 2;
    rng f1 c= the carrier of (TOP-REAL 2)|P1 by RELSET_1:12;
  then dom f1 = the carrier of I[01] & rng f1 c= the carrier of (TOP-REAL 2)|P
   by A16,FUNCT_2:def 1,XBOOLE_1:1;
  then f1 is Function of the carrier of I[01], the carrier of (TOP-REAL 2)|P'
       by FUNCT_2:def 1,RELSET_1:11;
  then reconsider ff1=f1 as map of I[01], (TOP-REAL 2)|P';
    f1 is continuous & (TOP-REAL 2)|P1' is SubSpace of (TOP-REAL 2)|P'
   by A5,A16,TOPMETR:4,TOPS_2:def 5;
then A17: ff1 is continuous by TOPMETR:7;
   [#]((TOP-REAL 2)|P2) = P2 & [#]((TOP-REAL 2)|P) = P & P2 c= P
    by A3,PRE_TOPC:def 10,XBOOLE_1:7;
  then the carrier of ((TOP-REAL 2)|P2) c= [#]((TOP-REAL 2)|P) by PRE_TOPC:12;
  then A18: the carrier of ((TOP-REAL 2)|P2) c= the carrier of ((TOP-REAL 2)|P)
   by PRE_TOPC:12;
    rng f2 c= the carrier of (TOP-REAL 2)|P2 by RELSET_1:12;
  then dom f2 = the carrier of I[01] & rng f2 c= the carrier of (TOP-REAL 2)|P
   by A18,FUNCT_2:def 1,XBOOLE_1:1;
  then f2 is Function of the carrier of I[01], the carrier of (TOP-REAL 2)|P'
     by FUNCT_2:def 1,RELSET_1:11;
  then reconsider ff2=f2 as map of I[01], (TOP-REAL 2)|P';
    f2 is continuous & (TOP-REAL 2)|P2' is SubSpace of (TOP-REAL 2)|P'
   by A7,A18,TOPMETR:4,TOPS_2:def 5;
then A19: ff2 is continuous by TOPMETR:7;
  set k1 = ff1*(g1"), k2 = ff2*(g2");
  reconsider RS = R^2-unit_square
    as non empty Subset of TOP-REAL 2;
    len h1 >= 2 & len h2 >= 2 by A9,TOPREAL1:def 10;
  then reconsider Lh1 = L~h1, Lh2 = L~h2
   as non empty Subset of TOP-REAL 2 by TOPREAL1:29;
  reconsider g1 as map of I[01], (TOP-REAL 2)|Lh1;
  reconsider g2 as map of I[01], (TOP-REAL 2)|Lh2;
  set T1 = (TOP-REAL 2)|Lh1, T2 = (TOP-REAL 2)|Lh2,
      T = (TOP-REAL 2)|RS;
    R^2-unit_square = [#]
(T) by PRE_TOPC:def 10 .= the carrier of T by PRE_TOPC:12;
  then reconsider p00,p11 as Point of T by Lm21,Lm22,TOPREAL1:def 3;
A20: [#] T1 = L~h1 & [#] T2 = L~h2 & [#]
 T = R^2-unit_square by PRE_TOPC:def 10;
A21: I[01] is compact by HEINE:11,TOPMETR:27;
A22: g1 is continuous & g2 is continuous & rng g1 = [#](T1) & rng g2 = [#](T2)
     & dom g1 = [#]I[01] & dom g2 = [#]I[01] by A13,A14,TOPS_2:def 5;
then A23: T1 is compact & T2 is compact by A21,COMPTS_1:23;
      TopSpaceMetr(Euclid 2) is_T2 by PCOMPS_1:38;
    then A24: TOP-REAL 2 is_T2 by EUCLID:def 8;
then A25: T is_T2 by TOPMETR:3;
      g1" is continuous & g2" is continuous by A13,A14,TOPS_2:def 5;
then A26: k1 is continuous & k2 is continuous by A17,A19,TOPS_2:58;
A27: g1 is one-to-one & g2 is one-to-one by A13,A14,TOPS_2:def 5;
then A28: g2" = (g2 qua Function)" by A22,TOPS_2:def 4;
A29: g1" = (g1 qua Function)" by A22,A27,TOPS_2:def 4;
A30: dom g1 = the carrier of I[01] & the carrier of I[01] = [.0,1.]
& dom g2 = the carrier of I[01] &
      dom ff1 = the carrier of I[01] & dom ff2 = the carrier of I[01]
       by BORSUK_1:83,FUNCT_2:def 1;
then A31: 0 in dom g1 & 0 in dom g2 by RCOMP_1:15;
A32: 0 in dom ff2 & 0 in dom ff1 by A30,RCOMP_1:15;
A33: p00 in rng g2 by A14,A31,FUNCT_1:def 5;
A34: dom(g2") = rng g2 by A27,A28,FUNCT_1:54;
A35: p00 in dom (g2") by A27,A28,A33,FUNCT_1:54;
A36: 0 = (g2").p00 by A14,A27,A28,A31,FUNCT_1:54;
      (g2").p00 in dom ff2 by A14,A27,A28,A30,A31,FUNCT_1:54;
then A37: p00 in dom(ff2*(g2")) by A35,FUNCT_1:21;
A38: dom(g1") = rng g1 by A27,A29,FUNCT_1:54;
then A39: p00 in dom (g1") by A13,A31,FUNCT_1:def 5;
A40: 0 = (g1").p00 by A13,A27,A29,A31,FUNCT_1:54;
      (g1").p00 in dom ff1 by A13,A27,A29,A30,A31,FUNCT_1:54;
    then p00 in dom(ff1*(g1")) by A39,FUNCT_1:21;
then A41: k1.p00 = ff1.((g1").p00) by FUNCT_1:22
          .= p1 by A6,A13,A27,A29,A31,FUNCT_1:54;
then A42: k1.p00 = ff2.((g2").p00) by A8,A14,A27,A28,A31,FUNCT_1:54
          .= k2.p00 by A37,FUNCT_1:22;
A43: 1 in dom g1 & 1 in dom g2 by A30,RCOMP_1:15;
A44: 1 in dom ff2 & 1 in dom ff1 by A30,RCOMP_1:15;
A45: p11 in dom (g2") & p11 in dom (g1") by A13,A14,A34,A38,A43,FUNCT_1:def 5;
A46: 1 = (g2").p11 by A14,A27,A28,A43,FUNCT_1:54;
      (g2").p11 in dom ff2 by A14,A27,A28,A30,A43,FUNCT_1:54;
then A47: p11 in dom(ff2*(g2")) by A45,FUNCT_1:21;
A48: 1 = (g1").p11 by A13,A27,A29,A43,FUNCT_1:54;
      (g1").p11 in dom ff1 by A13,A27,A29,A30,A43,FUNCT_1:54;
    then p11 in dom(ff1*(g1")) by A45,FUNCT_1:21;
then A49: k1.p11 = ff1.((g1").p11) by FUNCT_1:22
           .= p2 by A6,A13,A27,A29,A43,FUNCT_1:54;
then A50: k1.p11 = ff2.((g2").p11) by A8,A14,A27,A28,A43,FUNCT_1:54
           .= k2.p11 by A47,FUNCT_1:22;
    [#] T1 c= [#] T & [#] T2 c= [#] T by A10,A20,XBOOLE_1:7;
  then the carrier of T1 c= [#] T & the carrier of T2 c= [#] T by PRE_TOPC:12;
  then the carrier of T1 c= the carrier of T & the carrier of T2 c=
   the carrier of T by PRE_TOPC:12;
  then T1 is SubSpace of T & T2 is SubSpace of T by TOPMETR:4;
  then consider h being map of T,(TOP-REAL 2)|P such that
A51: h = k1+*k2 & h is continuous by A10,A20,A23,A25,A26,A42,A50,TOPMETR2:5;
A52: T is compact by Th2,COMPTS_1:12;
A53: (TOP-REAL 2)|P' is_T2 by A24,TOPMETR:3;
A54: f1 is one-to-one by A5,TOPS_2:def 5;
A55: g1" is one-to-one by A27,A29,FUNCT_1:62;
then A56: k1 is one-to-one by A54,FUNCT_1:46;
A57: f2 is one-to-one by A7,TOPS_2:def 5;
A58: g2" is one-to-one by A27,A28,FUNCT_1:62;
then A59: k2 is one-to-one by A57,FUNCT_1:46;
A60: rng(g1") = dom g1 by A27,A29,FUNCT_1:55;
A61: rng(g2") = dom g2 by A27,A28,FUNCT_1:55;
then A62: dom k2 = dom(g2") by A30,RELAT_1:46;
A63:    k1.:(dom k1 /\ dom k2) c= rng k2
     proof
      let a be set; assume a in k1.:(dom k1 /\ dom k2);
      then A64: ex x being set st x in dom k1 & x in dom k1 /\ dom k2 &
 a = k1.x by FUNCT_1:def 12;
        dom k1 = the carrier of T1 & dom k2 = the carrier of T2 &
       the carrier of T1 = [#] T1 & the carrier of T2 = [#] T2
        by FUNCT_2:def 1,PRE_TOPC:12;
      then a = p1 or a = p2 by A10,A20,A41,A49,A64,TARSKI:def 2;
      hence a in rng k2 by A35,A41,A42,A45,A49,A50,A62,FUNCT_1:def 5;
     end;
      dom f1 = the carrier of I[01] & dom g1 = the carrier of I[01]
     by FUNCT_2:def 1;
then A65: rng k1 = rng f1 by A60,RELAT_1:47 .= P1 by A5,A15,TOPS_2:def 5;
A66: rng k2 = rng f2 by A30,A61,RELAT_1:47 .= [#]
((TOP-REAL 2)|P2) by A7,TOPS_2:def 5
          .= P2 by PRE_TOPC:def 10;
then A67: rng h = [#]((TOP-REAL 2)|P') by A3,A15,A51,A63,A65,TOPMETR2:3;
      dom h = the carrier of T by FUNCT_2:def 1;
then A68: dom h = [#]T by PRE_TOPC:12;
      now let x1,x2 be set;
     assume that A69: x1 in dom k2 and A70: x2 in dom k1 \ dom k2;
     assume A71: k2.x1 = k1.x2;
     A72: x2 in dom k1 by A70,XBOOLE_0:def 4;
     then k2.x1 in P2 & k2.x1 in P1 by A65,A66,A69,A71,FUNCT_1:def 5;
     then A73: k2.x1 in P1 /\ P2 by XBOOLE_0:def 3;

    A74: x1 in dom(g2") by A69,FUNCT_1:21;
    A75: x2 in dom(g1") by A72,FUNCT_1:21;
      per cases by A4,A73,TARSKI:def 2;
       suppose A76: k2.x1 = p1;
        then p1 = ff2.((g2").x1) & (g2").x1 in dom ff2 & ff2.0=p1
         by A8,A69,FUNCT_1:21,22;
        then (g2").x1 = 0 & p00 in
 dom (g2") by A27,A28,A32,A33,A57,FUNCT_1:54,def 8;
        then A77: x1 = p00 by A36,A58,A74,FUNCT_1:def 8;
          p1 = ff1.((g1").x2) & (g1").x2 in dom ff1 & ff1.0 = p1
         by A6,A71,A72,A76,FUNCT_1:21,22;
        then (g1").x2 = 0 & p00 in
 dom(g1") by A13,A30,A32,A38,A54,FUNCT_1:def 5,def 8;
        then x2 in dom k2 by A40,A55,A69,A75,A77,FUNCT_1:def 8;
       hence contradiction by A70,XBOOLE_0:def 4;
       suppose A78: k2.x1 = p2;
        then p2 = ff2.((g2").x1) & (g2").x1 in dom ff2 & ff2.1 = p2
         by A8,A69,FUNCT_1:21,22;
        then (g2").x1 = 1 & p11 in
 dom (g2") by A14,A30,A34,A44,A57,FUNCT_1:def 5,def 8;
        then A79: x1 = p11 by A46,A58,A74,FUNCT_1:def 8;
          p2 = ff1.((g1").x2) & (g1").x2 in dom ff1 & ff1.1 = p2
         by A6,A71,A72,A78,FUNCT_1:21,22;
        then (g1").x2 = 1 & p11 in
 dom(g1") by A13,A30,A38,A44,A54,FUNCT_1:def 5,def 8;
        then x2 in dom k2 by A48,A55,A69,A75,A79,FUNCT_1:def 8;
       hence contradiction by A70,XBOOLE_0:def 4;
    end;
then A80: h is one-to-one by A51,A56,A59,TOPMETR2:2;
   reconsider h as map of ((TOP-REAL 2)|R^2-unit_square),(TOP-REAL 2)|P;
  take h;
  thus h is_homeomorphism by A51,A52,A53,A67,A68,A80,COMPTS_1:26;
end;

theorem Th5:
for P being non empty Subset of TOP-REAL 2 holds
P is_simple_closed_curve iff
 (ex p1,p2 st p1 <> p2 & p1 in P & p2 in P)
 & for p1,p2 st p1 <> p2 & p1 in P & p2 in P
    ex P1,P2 being non empty Subset of TOP-REAL 2
     st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
     P = P1 \/ P2 & P1 /\ P2 = {p1,p2}
proof
let P be non empty Subset of TOP-REAL 2;
thus P is_simple_closed_curve implies
 (ex p1,p2 st p1 <> p2 & p1 in P & p2 in P)
& for p1,p2 st p1 <> p2 & p1 in P & p2 in P
   ex P1,P2 being non empty Subset of TOP-REAL 2 st
   P1 is_an_arc_of p1,p2 &
   P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2}
 proof
  assume A1: P is_simple_closed_curve;
  then consider f being map of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P
    such that A2: f is_homeomorphism by Def1;
  thus ex p1,p2 st p1 <> p2 & p1 in P & p2 in P by A1,Th4;
  let p1,p2; assume that
A3: p1 <> p2 and A4: p1 in P and A5: p2 in P;
A6: f is one-to-one by A2,TOPS_2:def 5;
A7: f is continuous by A2,TOPS_2:def 5;
A8: rng f = [#]((TOP-REAL 2)|P) & dom f = [#]((TOP-REAL 2)|R^2-unit_square)
        by A2,TOPS_2:def 5;
then A9: f" = (f qua Function)" by A6,TOPS_2:def 4;
then A10: dom(f") = rng f by A6,FUNCT_1:54;
A11: f" is one-to-one by A6,A9,FUNCT_1:62;
A12: rng(f") = dom f by A6,A9,FUNCT_1:55;
A13: [#]((TOP-REAL 2)|P) = P by PRE_TOPC:def 10;
A14: p1 in dom(f") & p2 in dom(f") by A4,A5,A8,A10,PRE_TOPC:def 10;
   set q1 = (f").p1, q2 = (f").p2;
   set RS = R^2-unit_square;
  reconsider f as map of (TOP-REAL 2)|RS, (TOP-REAL 2)|P;
A15: [#]((TOP-REAL 2)|RS) c= [#](TOP-REAL 2) & rng(f") =
    [#]((TOP-REAL 2)|RS) & q1 in rng(f") & q2 in rng(f")
     by A12,A14,FUNCT_1:def 5,PRE_TOPC:def 9,TOPS_2:51;
    then reconsider q1, q2 as Point of TOP-REAL 2 by PRE_TOPC:12;
A16: q1 <> q2 by A3,A11,A14,FUNCT_1:def 8;
A17: dom f = the carrier of (TOP-REAL 2)|R^2-unit_square by FUNCT_2:def 1;
A18:the carrier of (TOP-REAL 2)|R^2-unit_square = [#]((TOP-REAL 2)|
R^2-unit_square)
     by PRE_TOPC:12;
A19: [#]((TOP-REAL 2)|R^2-unit_square) = R^2-unit_square by PRE_TOPC:def 10;
  then q1 in R^2-unit_square & q2 in R^2-unit_square
 by A12,A14,A17,A18,FUNCT_1:def 5;
  then consider Q1,Q2 being non empty Subset of TOP-REAL 2
  such that
A20: Q1 is_an_arc_of q1,q2 & Q2 is_an_arc_of q1,q2 and
A21: R^2-unit_square = Q1 \/ Q2 & Q1 /\ Q2 = {q1,q2} by A16,Th1;
  set P1 = f.:Q1, P2 = f.:Q2;
A22: Q1 c= dom f & Q2 c= dom f by A17,A18,A19,A21,XBOOLE_1:7;
    P1 c= rng f & P2 c= rng f & [#]((TOP-REAL 2)|P) c= [#](TOP-REAL 2)
   by PRE_TOPC:def 9,RELAT_1:144;
  then P1 c= [#](TOP-REAL 2) & P2 c= [#](TOP-REAL 2) & [#](TOP-REAL 2) =
   the carrier of TOP-REAL 2 by A8,PRE_TOPC:12,XBOOLE_1:1;
  then reconsider P1, P2 as non empty Subset of TOP-REAL 2 by
A22,RELAT_1:152;
  take P1,P2;
A23: rng(f|Q1) = P1 by RELAT_1:148 .= [#]((TOP-REAL 2)|P1) by PRE_TOPC:def 10
           .= the carrier of (TOP-REAL 2)|P1 by PRE_TOPC:12;
A24: dom(f|Q1) = R^2-unit_square /\ Q1 by A17,A18,A19,RELAT_1:90
           .= Q1 by A21,XBOOLE_1:21 .= [#]((TOP-REAL 2)|Q1) by PRE_TOPC:def 10;
  then dom(f|Q1) = the carrier of (TOP-REAL 2)|Q1 &
   rng(f|Q1) c= the carrier of (TOP-REAL 2)|P1 by A23,PRE_TOPC:12;
  then f|Q1 is Function of the carrier of (TOP-REAL 2)|Q1,
                           the carrier of (TOP-REAL 2)|P1
   by FUNCT_2:def 1,RELSET_1:11;
  then reconsider F1 = f|Q1 as map of (TOP-REAL 2)|Q1, (TOP-REAL 2)|P1;
    [#]((TOP-REAL 2)|Q1) = Q1 by PRE_TOPC:def 10;
  then [#]((TOP-REAL 2)|Q1) c= the carrier of (TOP-REAL 2)|R^2-unit_square
   by A18,A19,A21,XBOOLE_1:7;
   then the carrier of (TOP-REAL 2)|Q1 c=
  the carrier of (TOP-REAL 2)|R^2-unit_square by PRE_TOPC:12;
then A25: (TOP-REAL 2)|Q1 is SubSpace of (TOP-REAL 2)|R^2-unit_square by
TOPMETR:4;
A26: rng F1 = [#]((TOP-REAL 2)|P1) by A23,PRE_TOPC:12;
A27: F1 is one-to-one by A6,FUNCT_1:84;
      Q1 c= f"P1 & Q2 c= f"P2 & f"P1 c= Q1 & f"P2 c= Q2
     by A6,A22,FUNCT_1:146,152;
then A28:  f"P1 = Q1 & f"P2 = Q2 by XBOOLE_0:def 10;
       for R being Subset of (TOP-REAL 2)|P1
        st R is closed holds F1"R is closed
      proof
       let R be Subset of (TOP-REAL 2)|P1;
       assume R is closed;
       then consider S1 being Subset of TOP-REAL 2 such that
      A29: S1 is closed and A30: R = S1 /\ [#]((TOP-REAL 2)|P1) by PRE_TOPC:43;
        P /\ the carrier of TOP-REAL 2 = P & S1 c= the carrier of TOP-REAL 2
       by XBOOLE_1:28;
      then rng f /\ S1 c= P by A8,A13,XBOOLE_1:26;
      then rng f /\ S1 is Subset of (TOP-REAL 2)|P
        by A13,PRE_TOPC:12;
      then reconsider S2 = rng f /\
 S1 as Subset of (TOP-REAL 2)|P;
      A31: F1"R = Q1 /\ (f"R) by FUNCT_1:139
              .= Q1 /\ ((f"S1) /\ f"([#]((TOP-REAL 2)|P1))) by A30,FUNCT_1:137
              .= (f"S1) /\ Q1 /\ Q1 by A28,PRE_TOPC:def 10
              .= (f"S1) /\ (Q1 /\ Q1) by XBOOLE_1:16
              .= (f"S1) /\ [#]((TOP-REAL 2)|Q1) by PRE_TOPC:def 10
              .= (f"(rng f /\ S1)) /\ [#]((TOP-REAL 2)|Q1) by RELAT_1:168;
         S2 is closed by A8,A29,PRE_TOPC:43;
       then f"S2 is closed by A7,PRE_TOPC:def 12;
       hence F1"R is closed by A25,A31,PRE_TOPC:43;
      end;
then A32: F1 is continuous by PRE_TOPC:def 12;
    consider ff being map of I[01], (TOP-REAL 2)|Q1 such that
     A33: ff is_homeomorphism and ff.0 = q1 & ff.1 = q2 by A20,TOPREAL1:def 2;
A34: I[01] is compact by HEINE:11,TOPMETR:27;
    reconsider Q1'=Q1, Q2'=Q2
     as non empty Subset of TOP-REAL 2;
      ff is continuous & rng ff = [#]((TOP-REAL 2)|Q1) by A33,TOPS_2:def 5;
then A35: (TOP-REAL 2)|Q1' is compact by A34,COMPTS_1:23;
      TopSpaceMetr(Euclid 2) is_T2 by PCOMPS_1:38;
    then TOP-REAL 2 is_T2 by EUCLID:def 8;
 then (TOP-REAL 2)|P1 is_T2 by TOPMETR:3;
then A36: F1 is_homeomorphism by A24,A26,A27,A32,A35,COMPTS_1:26;
      q1 in {q1,q2} & {q1,q2} c= Q1 by A21,TARSKI:def 2,XBOOLE_1:17;
    then A37: q1 in dom f /\ Q1 by A12,A15,XBOOLE_0:def 3;
A38: p1 = f.q1 by A6,A9,A10,A14,FUNCT_1:57 .= F1.q1 by A37,FUNCT_1:71;
      q2 in {q1,q2} & {q1,q2} c= Q1 by A21,TARSKI:def 2,XBOOLE_1:17;
    then A39: q2 in dom f /\ Q1 by A12,A15,XBOOLE_0:def 3;
    p2 = f.q2 by A6,A9,A10,A14,FUNCT_1:57 .= F1.q2 by A39,FUNCT_1:71;
  hence P1 is_an_arc_of p1,p2 by A20,A36,A38,Th3;
A40: rng(f|Q2) = P2 by RELAT_1:148
           .= [#]((TOP-REAL 2)|P2) by PRE_TOPC:def 10
           .= the carrier of (TOP-REAL 2)|P2 by PRE_TOPC:12;
A41: dom(f|Q2) = R^2-unit_square /\ Q2 by A17,A18,A19,RELAT_1:90
           .= Q2 by A21,XBOOLE_1:21 .= [#]((TOP-REAL 2)|Q2) by PRE_TOPC:def 10;
  then dom(f|Q2) = the carrier of (TOP-REAL 2)|Q2 &
   rng(f|Q2) c= the carrier of (TOP-REAL 2)|P2 by A40,PRE_TOPC:12;
  then f|Q2 is Function of the carrier of (TOP-REAL 2)|Q2,
                           the carrier of (TOP-REAL 2)|P2
   by FUNCT_2:def 1,RELSET_1:11;
  then reconsider F2 = f|Q2 as map of (TOP-REAL 2)|Q2, (TOP-REAL 2)|P2;
    [#]((TOP-REAL 2)|Q2) = Q2 by PRE_TOPC:def 10;
  then [#]((TOP-REAL 2)|Q2) c= the carrier of (TOP-REAL 2)|R^2-unit_square
   by A18,A19,A21,XBOOLE_1:7;
   then the carrier of (TOP-REAL 2)|Q2 c=
  the carrier of (TOP-REAL 2)|R^2-unit_square by PRE_TOPC:12;
then A42: (TOP-REAL 2)|Q2 is SubSpace of (TOP-REAL 2)|R^2-unit_square by
TOPMETR:4;
A43: rng F2 = [#]((TOP-REAL 2)|P2) by A40,PRE_TOPC:12;
A44: F2 is one-to-one by A6,FUNCT_1:84;
       for R being Subset of (TOP-REAL 2)|P2 st R is closed holds F2"R is
closed
      proof
       let R be Subset of (TOP-REAL 2)|P2;
       assume R is closed;
       then consider S1 being Subset of TOP-REAL 2 such that
      A45: S1 is closed and A46: R = S1 /\ [#]((TOP-REAL 2)|P2) by PRE_TOPC:43;
        P /\ the carrier of TOP-REAL 2 = P & S1 c= the carrier of TOP-REAL 2
       by XBOOLE_1:28;
      then rng f /\ S1 c= P by A8,A13,XBOOLE_1:26;
      then rng f /\ S1 is Subset of (TOP-REAL 2)|P
        by A13,PRE_TOPC:12;
      then reconsider S2 = rng f /\
 S1 as Subset of (TOP-REAL 2)|P;
      A47: F2"R = Q2 /\ (f"R) by FUNCT_1:139
              .= Q2 /\ ((f"S1) /\ f"([#]((TOP-REAL 2)|P2))) by A46,FUNCT_1:137
              .= (f"S1) /\ Q2 /\ Q2 by A28,PRE_TOPC:def 10
              .= (f"S1) /\ (Q2 /\ Q2) by XBOOLE_1:16
              .= (f"S1) /\ [#]((TOP-REAL 2)|Q2) by PRE_TOPC:def 10
              .= (f"(rng f /\ S1)) /\ [#]((TOP-REAL 2)|Q2) by RELAT_1:168;
         S2 is closed by A8,A45,PRE_TOPC:43;
       then f"S2 is closed by A7,PRE_TOPC:def 12;
       hence F2"R is closed by A42,A47,PRE_TOPC:43;
      end;
then A48: F2 is continuous by PRE_TOPC:def 12;
    consider ff being map of I[01], (TOP-REAL 2)|Q2 such that
A49: ff is_homeomorphism and ff.0 = q1 & ff.1 = q2 by A20,TOPREAL1:def 2;
      ff is continuous & rng ff = [#]((TOP-REAL 2)|Q2) by A49,TOPS_2:def 5;
then A50: (TOP-REAL 2)|Q2' is compact by A34,COMPTS_1:23;
      TopSpaceMetr(Euclid 2) is_T2 by PCOMPS_1:38;
    then TOP-REAL 2 is_T2 by EUCLID:def 8;
 then (TOP-REAL 2)|P2 is_T2 by TOPMETR:3;
then A51: F2 is_homeomorphism by A41,A43,A44,A48,A50,COMPTS_1:26;
    q1 in {q1,q2} & {q1,q2} c= Q2 by A21,TARSKI:def 2,XBOOLE_1:17;
  then A52: q1 in dom f /\ Q2 by A12,A15,XBOOLE_0:def 3;
A53: p1 = f.q1 by A6,A9,A10,A14,FUNCT_1:57;
then A54: p1 = F2.q1 by A52,FUNCT_1:71;
    q2 in {q1,q2} & {q1,q2} c= Q2 by A21,TARSKI:def 2,XBOOLE_1:17;
  then A55: q2 in dom f /\ Q2 by A12,A15,XBOOLE_0:def 3;
A56: p2 = f.q2 by A6,A9,A10,A14,FUNCT_1:57;
   then p2 = F2.q2 by A55,FUNCT_1:71;
  hence P2 is_an_arc_of p1,p2 by A20,A51,A54,Th3;
  thus P = f.:(Q1 \/ Q2) by A8,A13,A19,A21,RELAT_1:146
        .= P1 \/ P2 by RELAT_1:153;
  thus P1 /\ P2 = f.:(Q1 /\ Q2) by A6,FUNCT_1:121
              .= f.:({q1} \/ {q2}) by A21,ENUMSET1:41
              .= f.:({q1}) \/ f.:({q2}) by RELAT_1:153
              .= {p1} \/ f.:({q2}) by A12,A15,A53,FUNCT_1:117
              .= {p1} \/ {p2} by A12,A15,A56,FUNCT_1:117
              .= {p1,p2} by ENUMSET1:41;
 end;
 given p1,p2 such that
 A57: p1 <> p2 & p1 in P & p2 in P;
 assume for p1,p2 st p1 <> p2 & p1 in P & p2 in P
   ex P1,P2 being non empty Subset of TOP-REAL 2
   st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
     P = P1 \/ P2 & P1 /\ P2 = {p1,p2};
 then ex P1,P2 being non empty Subset of TOP-REAL 2
 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
   P = P1 \/ P2 & P1 /\ P2 = {p1,p2} by A57;
 hence thesis by A57,Lm27;
end;

theorem
  for P being non empty Subset of TOP-REAL 2 holds
P is_simple_closed_curve iff
ex p1,p2 being Point of TOP-REAL 2,
   P1,P2 being non empty Subset of TOP-REAL 2
  st p1 <> p2 & p1 in P & p2 in P &
    P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
    P = P1 \/ P2 & P1 /\ P2 = {p1,p2}
proof
let P be non empty Subset of TOP-REAL 2;
 hereby
  assume A1: P is_simple_closed_curve;
  then consider p1,p2 such that
A2:  p1 <> p2 & p1 in P & p2 in P by Th5;
  consider P1,P2 being non empty Subset of TOP-REAL 2 such that
A3:  P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
      P = P1 \/ P2 & P1 /\ P2 = {p1,p2} by A1,A2,Th5;
  take p1,p2,P1,P2;
  thus p1 <> p2 & p1 in P & p2 in P &
       P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 &
        P = P1 \/ P2 & P1 /\ P2 = {p1,p2} by A2,A3;
 end;
 thus thesis by Lm27;
end;

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