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
  for A, B being Subset of TOP-REAL 2 st A c= B & W-bound A + E-bound A
  = W-bound B + E-bound B & A /\ Vertical_Line ((W-bound A + E-bound A) / 2) is
  non empty & proj2.:(B /\ Vertical_Line ((W-bound A + E-bound A) / 2)) is
  bounded_above holds (UMP A)`2 <= (UMP B)`2
proof
  let A, B be Subset of TOP-REAL 2;
  assume that
A1: A c= B and
A2: W-bound A + E-bound A = W-bound B + E-bound B and
A3: A /\ Vertical_Line ((W-bound A + E-bound A) / 2) is non empty and
A4: proj2.:(B /\ Vertical_Line ((W-bound A + E-bound A) / 2)) is bounded_above;
  set w = (W-bound A + E-bound A) / 2;
  proj2.:(A /\ Vertical_Line w) is non empty & A /\ Vertical_Line w c= B
  /\ Vertical_Line w by A1,A3,Lm1,RELAT_1:119,XBOOLE_1:26;
  then
  upper_bound(proj2.:(A /\ Vertical_Line w)) <=
   upper_bound(proj2.:(B /\ Vertical_Line w))
  by A4,RELAT_1:123,SEQ_4:48;
  then (UMP A)`2 <= upper_bound(proj2.:(B /\ Vertical_Line w)) by EUCLID:52;
  hence thesis by A2,EUCLID:52;
end;
