reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem
  for S being Segmentation of C, p st p in C
  ex i being Nat st i in dom S & p in Segm(S,i)
proof
  let S be Segmentation of C, p such that
A1: p in C;
  set X = { i : i in dom S & LE S/.i, p, C };
A2: X c= dom S
  proof
    let e be object;
    assume e in X;
    then ex i st e = i & i in dom S & LE S/.i, p, C;
    hence thesis;
  end;
A3: 1 in dom S by FINSEQ_5:6;
A4: W-min C = S/.1 by Def3;
  then LE S/.1, p, C by A1,JORDAN7:3;
  then 1 in X by A3;
  then reconsider X as finite non empty Subset of NAT by A2,XBOOLE_1:1;
  reconsider mX = max X as Nat by TARSKI:1;
  take mX;
A5: max X in X by XXREAL_2:def 8;
  hence mX in dom S by A2;
A6: 1 <= max X by A2,A5,FINSEQ_3:25;
A7: max X <= len S by A2,A5,FINSEQ_3:25;
A8: ex i st max X = i & i in dom S & LE S/.i, p, C by A5;
  per cases by A7,XXREAL_0:1;
  suppose
A9: max X < len S;
A10: 1 <= max X + 1 by NAT_1:11;
    max X +1 <= len S by A9,NAT_1:13;
    then
A11: mX + 1 in dom S by A10,FINSEQ_3:25;
A12: S is one-to-one by Def3;
    max X +1 >= 1+1 by A6,XREAL_1:6;
    then max X + 1 <> 1;
    then
A13: S/.(max X+1) <> S/.1 by A3,A11,A12,PARTFUN2:10;
A14: S/.(max X+1) in rng S by A11,PARTFUN2:2;
A15: rng S c= C by Def3;
    now
      assume LE S/.(max X+1),p,C;
      then max X +1 in X by A11;
      then max X +1 <= max X by XXREAL_2:def 8;
      hence contradiction by XREAL_1:29;
    end;
    then LE p,S/.(max X+1),C by A1,A14,A15,JORDAN16:7;
    then p in {p1: LE S/.(max X),p1,C & LE p1,S/.(max X+1),C} by A8;
    then p in Segment(S/.max X,S/.(max X+1),C) by A4,A13,JORDAN7:def 1;
    hence thesis by A6,A9,Def4;
  end;
  suppose
A16: max X = len S;
    now per cases;
      case p<>W-min C;
        thus LE S/.len S,p,C by A8,A16;
      end;
      case p=W-min C;
A17:    S/.len S in rng S by FINSEQ_6:168;
        rng S c= C by Def3;
        hence S/.len S in C by A17;
      end;
    end;
    then p in {p1: LE S/.len S,p1,C or S/.len S in C & p1=W-min C};
    then p in Segment(S/.len S,S/.1,C) by A4,JORDAN7:def 1;
    hence thesis by A16,Def4;
  end;
end;
