reserve n for Nat;

theorem
  for X, Y being non empty compact Subset of TOP-REAL 2 st X c= Y & (
  N-min Y in X or N-max Y in X) holds N-bound X = N-bound Y
proof
  let X, Y be non empty compact Subset of TOP-REAL 2;
  assume that
A1: X c= Y and
A2: N-min Y in X or N-max Y in X;
A3: (N-max X)`2 = N-bound X by EUCLID:52;
A4: (N-max Y)`2 = N-bound Y by EUCLID:52;
A5: (N-min Y)`2 = N-bound Y by EUCLID:52;
  (N-min X)`2 = N-bound X by EUCLID:52;
  hence thesis by A1,A2,A3,A5,A4,Th15,Th16;
end;
