reserve x for set;
reserve a,b,d,ra,rb,r0,s1,s2 for Real;
reserve r,s,r1,r2,r3,rc for Real;
reserve p,q,q1,q2 for Point of TOP-REAL 2;
reserve X,Y,Z for non empty TopSpace;

theorem Th14:
  for P being non empty Subset of TOP-REAL 2 st P is
  being_simple_closed_curve holds not (ex r st for p st p in P holds p`2=r)
proof
  let P be non empty Subset of TOP-REAL 2;
  assume
A1: P is being_simple_closed_curve;
  now
A2: [.0,1.] = ].0,1.[ \/ {0,1} by XXREAL_1:128;
    given r0 such that
A3: for p st p in P holds p`2=r0;
    consider p1,p2 being Point of TOP-REAL 2, P1,P2 being non empty Subset of
    TOP-REAL 2 such that
A4: p1 <> p2 and
A5: p1 in P and
A6: p2 in P and
A7: P1 is_an_arc_of p1,p2 and
A8: P2 is_an_arc_of p1,p2 and
A9: P = P1 \/ P2 and
A10: P1 /\ P2 = {p1,p2} by A1,TOPREAL2:6;
A11: p1`2=r0 by A3,A5;
A12: p2`2=r0 by A3,A6;
    then
A13: p1`2=p2`2 by A3,A5;
A14: now
      assume p1`1=p2`1;
      then p1=|[p2`1,p2`2]| by A13,EUCLID:53;
      hence contradiction by A4,EUCLID:53;
    end;
    consider f2 being Function of I[01], (TOP-REAL 2)|P2 such that
A15: f2 is being_homeomorphism and
A16: f2.0 = p1 and
A17: f2.1 = p2 by A8,TOPREAL1:def 1;
    f2 is continuous by A15,TOPS_2:def 5;
    then consider g2 being Function of I[01],R^1 such that
A18: g2 is continuous and
A19: for r,q2 st r in the carrier of I[01] & q2= f2.r holds q2`1=g2.r by Th12;
A20: [#]((TOP-REAL 2)|P2) = P2 by PRE_TOPC:def 5;
    1 in {0,1} by TARSKI:def 2;
    then
A21: 1 in the carrier of I[01] by A2,BORSUK_1:40,XBOOLE_0:def 3;
    then
A22: p2`1=g2.1 by A17,A19;
    0 in {0,1} by TARSKI:def 2;
    then
A23: 0 in the carrier of I[01] by A2,BORSUK_1:40,XBOOLE_0:def 3;
    then p1`1=g2.0 by A16,A19;
    then consider r2 such that
A24: 0<r2 & r2<1 and
A25: g2.r2=(p1`1+p2`1)/2 by A18,A22,A14,Th9;
A26: [.0,1.] = {r3 : 0<=r3 & r3<=1 } by RCOMP_1:def 1;
    then
A27: r2 in the carrier of I[01] by A24,BORSUK_1:40;
    dom f2=the carrier of I[01] by FUNCT_2:def 1;
    then
A28: f2.r2 in rng f2 by A27,FUNCT_1:def 3;
    then
A29: f2.r2 in P by A9,A20,XBOOLE_0:def 3;
    f2.r2 in P2 by A28,A20;
    then reconsider p4=f2.r2 as Point of (TOP-REAL 2);
A30: [#]((TOP-REAL 2)|P1) = P1 by PRE_TOPC:def 5;
    consider f1 being Function of I[01], (TOP-REAL 2)|P1 such that
A31: f1 is being_homeomorphism and
A32: f1.0 = p1 and
A33: f1.1 = p2 by A7,TOPREAL1:def 1;
    f1 is continuous by A31,TOPS_2:def 5;
    then consider g1 being Function of I[01],R^1 such that
A34: g1 is continuous and
A35: for r,q1 st r in the carrier of I[01] & q1= f1.r holds q1`1=g1.r by Th12;
A36: p2`1=g1.1 by A33,A35,A21;
    p1`1=g1.0 by A32,A35,A23;
    then consider r1 such that
A37: 0<r1 & r1<1 and
A38: g1.r1=(p1`1+p2`1)/2 by A34,A36,A14,Th9;
A39: r1 in the carrier of I[01] by A37,A26,BORSUK_1:40;
    then r1 in dom f1 by FUNCT_2:def 1;
    then
A40: f1.r1 in rng f1 by FUNCT_1:def 3;
    then f1.r1 in P1 by A30;
    then reconsider p3=f1.r1 as Point of (TOP-REAL 2);
    f1.r1 in P by A9,A40,A30,XBOOLE_0:def 3;
    then
A41: p3`2=r0 by A3
      .=p4`2 by A3,A29;
    p3`1= (p1`1+p2`1)/2 by A35,A38,A39
      .=p4`1 by A19,A25,A27;
    then f1.r1=f2.r2 by A41,TOPREAL3:6;
    then
A42: f1.r1 in P1/\P2 by A40,A30,A28,A20,XBOOLE_0:def 4;
    now
      per cases by A10,A42,TARSKI:def 2;
      case
A43:    f1.r1=p1;
A44:    ((1/2)*p1+(1/2)*p2)`2= ((1/2)*p1)`2+((1/2)*p2)`2 by TOPREAL3:2
          .=(1/2)*(p1`2)+((1/2)*p2)`2 by TOPREAL3:4
          .=(1/2)*r0+(1/2)*r0 by A11,A12,TOPREAL3:4
          .=r0;
        p1`1=2"*(p1`1)+ (p2`1)/2 by A35,A38,A39,A43
          .=(2"*p1)`1+ 2"*(p2`1) by TOPREAL3:4
          .=(2"*p1)`1+ (2"*p2)`1 by TOPREAL3:4
          .=((1/2)*p1+(1/2)*p2)`1 by TOPREAL3:2;
        then p1=(1/2)*p1+(1/2)*p2 by A11,A44,TOPREAL3:6;
        then 1*p1-(1/2)*p1=(1/2)*p1+(1/2)*p2-(1/2)*p1 by RLVECT_1:def 8;
        then 1*p1-(1/2)*p1=(1/2)*p2+((1/2)*p1-(1/2)*p1) by RLVECT_1:def 3;
        then 1*p1-(1/2)*p1=(1/2)*p2+(0.TOP-REAL 2) by RLVECT_1:5;
        then 1*p1-(1/2)*p1=(1/2)*p2 by RLVECT_1:4;
        then (1-(1/2))*p1=(1/2)*p2 by RLVECT_1:35;
        hence contradiction by A4,RLVECT_1:36;
      end;
      case
A45:    f1.r1=p2;
A46:    ((1/2)*p1+(1/2)*p2)`2= ((1/2)*p1)`2+((1/2)*p2)`2 by TOPREAL3:2
          .=(1/2)*(p1`2)+((1/2)*p2)`2 by TOPREAL3:4
          .=(1/2)*r0+(1/2)*r0 by A11,A12,TOPREAL3:4
          .=r0;
        p2`1=2"*(p1`1)+ (p2`1)/2 by A35,A38,A39,A45
          .=(2"*p1)`1+ 2"*(p2`1) by TOPREAL3:4
          .=(2"*p1)`1+ (2"*p2)`1 by TOPREAL3:4
          .=((1/2)*p1+(1/2)*p2)`1 by TOPREAL3:2;
        then p2=(1/2)*p1+(1/2)*p2 by A12,A46,TOPREAL3:6;
        then 1*p2-(1/2)*p2=(1/2)*p1+(1/2)*p2-(1/2)*p2 by RLVECT_1:def 8;
        then 1*p2-(1/2)*p2=(1/2)*p1+((1/2)*p2-(1/2)*p2) by RLVECT_1:def 3;
        then 1*p2-(1/2)*p2=(1/2)*p1+(0.TOP-REAL 2) by RLVECT_1:5;
        then 1*p2-(1/2)*p2=(1/2)*p1 by RLVECT_1:4;
        then (1-(1/2))*p2=(1/2)*p1 by RLVECT_1:35;
        hence contradiction by A4,RLVECT_1:36;
      end;
    end;
    hence contradiction;
  end;
  hence thesis;
end;
