reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem Th48:
  for P,P1,P2,P19,P29 being Subset of TOP-REAL 2,
  p1,p2 being Point of TOP-REAL 2 st P is being_simple_closed_curve
  & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2=P
  & P19 is_an_arc_of p1,p2 & P29 is_an_arc_of p1,p2 & P19 \/ P29=P
  holds P1=P19 & P2=P29 or P1=P29 & P2=P19
proof
  let P,P1,P2,P19,P29 be Subset of
  the carrier of TOP-REAL 2,p1,p2 be Point of TOP-REAL 2;
  assume that
A1: P is being_simple_closed_curve and
A2: P1 is_an_arc_of p1,p2 and
A3: P2 is_an_arc_of p1,p2 and
A4: P1 \/ P2=P and
A5: P19 is_an_arc_of p1,p2 and
A6: P29 is_an_arc_of p1,p2 and
A7: P19 \/ P29=P;
  reconsider P as Simple_closed_curve by A1;
A8: p1<>p2 by A6,Th37;
A9: p1 in P19 by A5,TOPREAL1:1;
A10: p2 in P19 by A5,TOPREAL1:1;
A11: p1 in P2 by A3,TOPREAL1:1;
A12: p2 in P2 by A3,TOPREAL1:1;
A13: p1 in P1 by A2,TOPREAL1:1;
A14: p2 in P1 by A2,TOPREAL1:1;
A15: P1 c= P by A4,XBOOLE_1:7;
A16: [#]((TOP-REAL 2)|P)=P by PRE_TOPC:def 5;
  P1\{p1,p2} c= P1 by XBOOLE_1:36;
  then reconsider Q1=P1\{p1,p2} as Subset of (TOP-REAL 2)|P by A15,A16,
XBOOLE_1:1;
A17: P2 c= P by A4,XBOOLE_1:7;
  P2\{p1,p2} c= P2 by XBOOLE_1:36;
  then reconsider Q2=P2\{p1,p2} as Subset of (TOP-REAL 2)|P by A16,A17,
XBOOLE_1:1;
A18: P19 c= P by A7,XBOOLE_1:7;
  P19\{p1,p2} c= P19 by XBOOLE_1:36;
  then reconsider Q19=P19\{p1,p2} as Subset of (TOP-REAL 2)|P by A16,A18,
XBOOLE_1:1;
A19: P29 c= P by A7,XBOOLE_1:7;
  P29\{p1,p2} c= P29 by XBOOLE_1:36;
  then reconsider Q29=P29\{p1,p2} as Subset of (TOP-REAL 2)|P by A16,A19,
XBOOLE_1:1;
A20: Q1 \/ Q2 =P\{p1,p2} by A4,XBOOLE_1:42;
A21: Q19 \/ Q29 =P\{p1,p2} by A7,XBOOLE_1:42;
  then
A22: Q19 \/ (Q1 \/ Q2)=Q1 \/ Q2 by A20,XBOOLE_1:7,12;
A23: Q29 \/ (Q1 \/ Q2)=Q1 \/ Q2 by A20,A21,XBOOLE_1:7,12;
A24: Q1 \/ (Q19 \/ Q29)=Q19 \/ Q29 by A20,A21,XBOOLE_1:7,12;
A25: Q2 \/ (Q19 \/ Q29)=Q19 \/ Q29 by A20,A21,XBOOLE_1:7,12;
  [#]((TOP-REAL 2)|P1)=P1 by PRE_TOPC:def 5;
  then reconsider R1=Q1 as Subset of (TOP-REAL 2)|P1 by XBOOLE_1:36;
  R1 is connected by A2,Th40;
  then
A26: Q1 is connected by A4,Th41,XBOOLE_1:7;
  [#]((TOP-REAL 2)|P2)=P2 by PRE_TOPC:def 5;
  then reconsider R2=Q2 as Subset of (TOP-REAL 2)|P2 by XBOOLE_1:36;
  R2 is connected by A3,Th40;
  then
A27: Q2 is connected by A4,Th41,XBOOLE_1:7;
  [#]((TOP-REAL 2)|P19)=P19 by PRE_TOPC:def 5;
  then reconsider R19=Q19 as Subset of (TOP-REAL 2)|P19 by XBOOLE_1:36;
  R19 is connected by A5,Th40;
  then
A28: Q19 is connected by A7,Th41,XBOOLE_1:7;
  [#]((TOP-REAL 2)|P29)=P29 by PRE_TOPC:def 5;
  then reconsider R29=Q29 as Subset of (TOP-REAL 2)|P29 by XBOOLE_1:36;
  R29 is connected by A6,Th40;
  then
A29: Q29 is connected by A7,Th41,XBOOLE_1:7;
A30: {p1,p2} c= P1
  by A13,A14,TARSKI:def 2;
A31: {p1,p2} c= P2
  by A11,A12,TARSKI:def 2;
A32: {p1,p2} c= P19
  by A9,A10,TARSKI:def 2;
A33: {p1,p2} c= P29
  proof
    let x be object;
    assume x in {p1,p2};
    then x=p1 or x=p2 by TARSKI:def 2;
    hence thesis by A6,TOPREAL1:1;
  end;
A34: Q1 \/ {p1,p2}=P1 by A30,XBOOLE_1:45;
A35: Q2 \/ {p1,p2}=P2 by A31,XBOOLE_1:45;
A36: Q19 \/ {p1,p2}=P19 by A32,XBOOLE_1:45;
A37: Q29 \/ {p1,p2}=P29 by A33,XBOOLE_1:45;
  now
    assume
A38: not(P1=P29 & P2=P19);
    now per cases by A38;
      case
A39:    P1<>P29;
A40:    now
          assume that
A41:      Q1\Q29={} and
A42:      Q29\Q1={};
A43:      Q1 c= Q29 by A41,XBOOLE_1:37;
          Q29 c= Q1 by A42,XBOOLE_1:37;
          hence contradiction by A34,A37,A39,A43,XBOOLE_0:def 10;
        end;
        now per cases by A40;
          case
A44:        Q1\Q29<>{};
            set y = the Element of Q1\Q29;
A45:        y in Q1 by A44,XBOOLE_0:def 5;
            then
A46:        y in Q19 \/ Q29 by A20,A21,XBOOLE_0:def 3;
            not y in Q29 by A44,XBOOLE_0:def 5;
            then y in Q19 by A46,XBOOLE_0:def 3;
            then Q1 /\ Q19<>{} by A45,XBOOLE_0:def 4;
            then
A47:        Q1 meets Q19;
            now
              assume Q2 meets Q19;
              then Q1 \/ Q19 \/ Q2 is connected by A26,A27,A28,A47,JORDAN1:4;
              hence contradiction by A8,A13,A14,A15,A20,A22,Th47,XBOOLE_1:4;
            end;
            then
A48:        Q2/\Q19={};
A49:        Q2 c= Q29
            proof
              let x be object;
              assume
A50:          x in Q2;
              then x in Q1 \/ Q2 by XBOOLE_0:def 3;
              then x in Q19 or x in Q29 by A20,A21,XBOOLE_0:def 3;
              hence thesis by A48,A50,XBOOLE_0:def 4;
            end;
            Q19 c= Q1
            proof
              let x be object;
              assume
A51:          x in Q19;
              then x in Q1 \/ Q2 by A20,A21,XBOOLE_0:def 3;
              then x in Q1 or x in Q2 by XBOOLE_0:def 3;
              hence thesis by A48,A51,XBOOLE_0:def 4;
            end;
            hence thesis by A2,A3,A5,A6,A34,A35,A36,A37,A49,Th46,XBOOLE_1:9;
          end;
          case
A52:        Q29\Q1<>{};
            set y = the Element of Q29\Q1;
A53:        y in Q29 by A52,XBOOLE_0:def 5;
            then
A54:        y in Q2 \/ Q1 by A20,A21,XBOOLE_0:def 3;
            not y in Q1 by A52,XBOOLE_0:def 5;
            then y in Q2 by A54,XBOOLE_0:def 3;
            then Q29 /\ Q2<>{} by A53,XBOOLE_0:def 4;
            then
A55:        Q29 meets Q2;
            now
              assume Q19 meets Q2;
              then Q29 \/ Q2 \/ Q19 is connected by A27,A28,A29,A55,JORDAN1:4;
              hence contradiction by A8,A13,A14,A15,A21,A25,Th47,XBOOLE_1:4;
            end;
            then
A56:        Q19/\Q2={};
A57:        Q19 c= Q1
            proof
              let x be object;
              assume
A58:          x in Q19;
              then x in Q19 \/ Q29 by XBOOLE_0:def 3;
              then x in Q1 or x in Q2 by A20,A21,XBOOLE_0:def 3;
              hence thesis by A56,A58,XBOOLE_0:def 4;
            end;
            Q2 c= Q29
            proof
              let x be object;
              assume
A59:          x in Q2;
              then x in Q2 \/ Q1 by XBOOLE_0:def 3;
              then x in Q29 or x in Q19 by A20,A21,XBOOLE_0:def 3;
              hence thesis by A56,A59,XBOOLE_0:def 4;
            end;
            hence thesis by A2,A3,A5,A6,A34,A35,A36,A37,A57,Th46,XBOOLE_1:9;
          end;
        end;
        hence thesis;
      end;
      case
A60:    P2<>P19;
A61:    now
          assume that
A62:      Q2\Q19={} and
A63:      Q19\Q2={};
A64:      Q2 c= Q19 by A62,XBOOLE_1:37;
          Q19 c= Q2 by A63,XBOOLE_1:37;
          hence contradiction by A35,A36,A60,A64,XBOOLE_0:def 10;
        end;
        now per cases by A61;
          case
A65:        Q2\Q19<>{};
            set y = the Element of Q2\Q19;
A66:        y in Q2 by A65,XBOOLE_0:def 5;
            then
A67:        y in Q29 \/ Q19 by A20,A21,XBOOLE_0:def 3;
            not y in Q19 by A65,XBOOLE_0:def 5;
            then y in Q29 by A67,XBOOLE_0:def 3;
            then Q2 /\ Q29<>{} by A66,XBOOLE_0:def 4;
            then
A68:        Q2 meets Q29;
            now
              assume Q1 meets Q29;
              then Q2 \/ Q29 \/ Q1 is connected by A26,A27,A29,A68,JORDAN1:4;
              hence contradiction by A8,A13,A14,A15,A20,A23,Th47,XBOOLE_1:4;
            end;
            then
A69:        Q1/\Q29={};
A70:        Q1 c= Q19
            proof
              let x be object;
              assume
A71:          x in Q1;
              then x in Q2 \/ Q1 by XBOOLE_0:def 3;
              then x in Q29 or x in Q19 by A20,A21,XBOOLE_0:def 3;
              hence thesis by A69,A71,XBOOLE_0:def 4;
            end;
            Q29 c= Q2
            proof
              let x be object;
              assume
A72:          x in Q29;
              then x in Q2 \/ Q1 by A20,A21,XBOOLE_0:def 3;
              then x in Q2 or x in Q1 by XBOOLE_0:def 3;
              hence thesis by A69,A72,XBOOLE_0:def 4;
            end;
            hence thesis by A2,A3,A5,A6,A34,A35,A36,A37,A70,Th46,XBOOLE_1:9;
          end;
          case
A73:        Q19\Q2<>{};
            set y = the Element of Q19\Q2;
A74:        y in Q19 by A73,XBOOLE_0:def 5;
            then
A75:        y in Q1 \/ Q2 by A20,A21,XBOOLE_0:def 3;
            not y in Q2 by A73,XBOOLE_0:def 5;
            then y in Q1 by A75,XBOOLE_0:def 3;
            then Q19 /\ Q1<>{} by A74,XBOOLE_0:def 4;
            then
A76:        Q19 meets Q1;
            now
              assume Q29 meets Q1;
              then Q19 \/ Q1 \/ Q29 is connected by A26,A28,A29,A76,JORDAN1:4;
              hence contradiction by A8,A13,A14,A15,A21,A24,Th47,XBOOLE_1:4;
            end;
            then
A77:        Q29 /\ Q1 = {};
A78:        Q29 c= Q2
            proof
              let x be object;
              assume
A79:          x in Q29;
              then x in Q29 \/ Q19 by XBOOLE_0:def 3;
              then x in Q2 or x in Q1 by A20,A21,XBOOLE_0:def 3;
              hence thesis by A77,A79,XBOOLE_0:def 4;
            end;
            Q1 c= Q19
            proof
              let x be object;
              assume
A80:          x in Q1;
              then x in Q19 \/ Q29 by A20,A21,XBOOLE_0:def 3;
              then x in Q19 or x in Q29 by XBOOLE_0:def 3;
              hence thesis by A77,A80,XBOOLE_0:def 4;
            end;
            hence thesis by A2,A3,A5,A6,A34,A35,A36,A37,A78,Th46,XBOOLE_1:9;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
