reserve n for Nat;

theorem
  for X, Y being non empty compact Subset of TOP-REAL 2 st X c= Y & (
  S-min Y in X or S-max Y in X) holds S-bound X = S-bound Y
proof
  let X, Y be non empty compact Subset of TOP-REAL 2;
  assume that
A1: X c= Y and
A2: S-min Y in X or S-max Y in X;
A3: (S-max X)`2 = S-bound X by EUCLID:52;
A4: (S-max Y)`2 = S-bound Y by EUCLID:52;
A5: (S-min Y)`2 = S-bound Y by EUCLID:52;
  (S-min X)`2 = S-bound X by EUCLID:52;
  hence thesis by A1,A2,A3,A5,A4,Th19,Th20;
end;
