
theorem
  for L being antisymmetric transitive RelStr, X,Y being set st
  ex_sup_of X,L & ex_sup_of Y,L & ex_sup_of X \/ Y, L holds "\/"(X \/ Y, L) =
  "\/"(X,L)"\/""\/"(Y,L)
proof
  let L be antisymmetric transitive RelStr;
  let X,Y be set such that
A1: ex_sup_of X,L & ex_sup_of Y,L and
A2: ex_sup_of X \/ Y, L;
  set a = "\/"(X,L), b = "\/"(Y,L), c = "\/"(X \/ Y, L);
A3: a is_>=_than X & b is_>=_than Y by A1,Th30;
A4: now
    let d be Element of L;
    assume d >= a & d >= b;
    then d is_>=_than X & d is_>=_than Y by A3,Th4;
    then d is_>=_than X \/ Y by Th10;
    hence c <= d by A2,Th30;
  end;
  c >= a & c >= b by A1,A2,Th34,XBOOLE_1:7;
  hence thesis by A4,Th18;
end;
