
theorem Th34:
  for L being RelStr, X,Y being set st X c= Y & ex_sup_of X,L &
  ex_sup_of Y,L holds "\/"(X,L) <= "\/"(Y,L)
proof
  let L be RelStr, X,Y be set;
  assume that
A1: X c= Y and
A2: ex_sup_of X,L and
A3: ex_sup_of Y,L;
  "\/"(Y,L) is_>=_than X
  proof
    let x be Element of L;
    assume
A4: x in X;
    "\/"(Y,L) is_>=_than Y by A3,Def9;
    hence thesis by A1,A4;
  end;
  hence thesis by A2,Def9;
end;
