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 Th30:
  UMP D in D
proof
  set w = (W-bound D + E-bound D) / 2;
  set X = D /\ Vertical_Line w;
  D meets Vertical_Line w by Def1;
  then
A1: X is non empty;
  proj2.:X is closed & proj2.:X is bounded_above by Th13;
  then upper_bound (proj2.:X) in proj2.:X by A1,Lm1,RCOMP_1:12,RELAT_1:119;
  then consider x being Point of TOP-REAL 2 such that
A2: x in X and
A3: upper_bound (proj2.:X) = proj2.x by Lm2;
  x in Vertical_Line w by A2,XBOOLE_0:def 4;
  then
A4: x`1 = w by JORDAN6:31
    .= (UMP D)`1 by EUCLID:52;
  x`2 = upper_bound (proj2.:X) by A3,PSCOMP_1:def 6
    .= (UMP D)`2 by EUCLID:52;
  then x = UMP D by A4,TOPREAL3:6;
  hence thesis by A2,XBOOLE_0:def 4;
end;
