reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th34:
  LSeg(UMP D, |[ (W-bound D + E-bound D) / 2, N-bound D]|) /\ D = { UMP D }
proof
  set C = D;
  set w = (W-bound C + E-bound C) / 2;
  set L = LSeg(UMP C, |[w,N-bound C]|);
  set X = C /\ Vertical_Line(w);
A1: UMP C in C by Th30;
A2: UMP C in L by RLTOPSP1:68;
  hereby
    let x be object;
A3: (UMP C)`1 = w by EUCLID:52;
    assume
A4: x in L /\ C;
    then
A5: x in L by XBOOLE_0:def 4;
    reconsider y = x as Point of TOP-REAL 2 by A4;
    UMP C in C by Th30;
    then |[w,N-bound C]|`2 = N-bound C & (UMP C)`2 <= N-bound C by EUCLID:52
,PSCOMP_1:24;
    then
A6: (UMP C)`2 <= y`2 by A5,TOPREAL1:4;
A7: proj2.:X is bounded_above by Th13;
A8: (UMP C)`2 = upper_bound (proj2.:X) by EUCLID:52;
A9: x in C by A4,XBOOLE_0:def 4;
    L is vertical by Th32;
    then
A10: y`1 = w by A2,A5,A3,SPPOL_1:def 3;
    then y in Vertical_Line (w) by JORDAN6:31;
    then y`2 = proj2.y & y in X by A9,PSCOMP_1:def 6,XBOOLE_0:def 4;
    then y`2 in proj2.:X by FUNCT_2:35;
    then y`2 <= upper_bound (proj2.:X) by A7,SEQ_4:def 1;
    then y`2 = upper_bound (proj2.:X) by A8,A6,XXREAL_0:1;
    then y = UMP C by A3,A8,A10,TOPREAL3:6;
    hence x in {UMP C} by TARSKI:def 1;
  end;
  let x be object;
  assume x in {UMP C};
  then x = UMP C by TARSKI:def 1;
  hence thesis by A2,A1,XBOOLE_0:def 4;
end;
