reserve n for Nat;

theorem
  for X, Y being non empty compact Subset of TOP-REAL 2 st X c= Y & (
  E-min Y in X or E-max Y in X) holds E-bound X = E-bound Y
proof
  let X, Y be non empty compact Subset of TOP-REAL 2;
  assume that
A1: X c= Y and
A2: E-min Y in X or E-max Y in X;
A3: (E-max X)`1 = E-bound X by EUCLID:52;
A4: (E-max Y)`1 = E-bound Y by EUCLID:52;
A5: (E-min Y)`1 = E-bound Y by EUCLID:52;
  (E-min X)`1 = E-bound X by EUCLID:52;
  hence thesis by A1,A2,A3,A5,A4,Th17,Th18;
end;
