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 & UMP B in A & A /\
  Vertical_Line ((W-bound A + E-bound A) / 2) is non empty & proj2.:(B /\
  Vertical_Line ((W-bound B + E-bound B) / 2)) is bounded_above & W-bound A +
  E-bound A = W-bound B + E-bound B holds UMP A = UMP B
proof
  let A, B be Subset of TOP-REAL 2 such that
A1: A c= B and
A2: UMP B in A and
A3: A /\ Vertical_Line ((W-bound A + E-bound A) / 2) is non empty and
A4: proj2.:(B /\ Vertical_Line ((W-bound B + E-bound B) / 2)) is
  bounded_above and
A5: W-bound A + E-bound A = W-bound B + E-bound B;
  set w = (W-bound A + E-bound A)/2;
A6: (UMP A)`2 = upper_bound (proj2.:(A /\ Vertical_Line w)) & proj2.:(A /\
  Vertical_Line w) is non empty by A3,Lm1,EUCLID:52,RELAT_1:119;
A7: (UMP B)`1 = w by A5,EUCLID:52;
A8: for s being Real st 0 < s ex r being Real st r in proj2.:
  (A /\ Vertical_Line w) & (UMP B)`2 - s < r
  proof
    let s be Real;
    assume
A9: 0 < s;
    take (UMP B)`2;
    UMP B in Vertical_Line w by A7,JORDAN6:31;
    then proj2.UMP B = (UMP B)`2 & UMP B in A /\ Vertical_Line w by A2,
PSCOMP_1:def 6,XBOOLE_0:def 4;
    hence (UMP B)`2 in proj2.:(A /\ Vertical_Line w) by FUNCT_2:35;
    (UMP B)`2 - s < (UMP B)`2 - 0 by A9,XREAL_1:15;
    hence thesis;
  end;
A10: A /\ Vertical_Line w c= B /\ Vertical_Line w by A1,XBOOLE_1:26;
  then
A11: proj2.:(A /\ Vertical_Line w) c= proj2.:(B /\ Vertical_Line w) by
RELAT_1:123;
  (UMP B)`2 = upper_bound (proj2.:(B /\ Vertical_Line w)) by A5,EUCLID:52;
  then
A12: for r being Real st r in proj2.:(A /\ Vertical_Line w) holds (
  UMP B)`2 >= r by A4,A5,A11,SEQ_4:def 1;
  proj2.:(A /\ Vertical_Line w) is bounded_above by A4,A5,A10,RELAT_1:123
,XXREAL_2:43;
  then (UMP A)`1 = w & (UMP A)`2 = (UMP B)`2 by A6,A12,A8,EUCLID:52,SEQ_4:def 1
;
  hence thesis by A7,TOPREAL3:6;
end;
