theorem
  for X being non empty real-membered set, Y being real-membered set st
  X c= Y & Y is bounded_above holds upper_bound X <= upper_bound Y
proof
  let X be non empty real-membered set, Y be real-membered set;
  assume X c= Y & Y is bounded_above;
  then t in X implies t <= upper_bound Y by Def1;
  hence thesis by Th45;
end;
