reserve X,Y,x for set;

theorem Th1:
  for A being set holds A is preBoolean iff for X,Y being set st X
  in A & Y in A holds X \/ Y in A & X \ Y in A
proof
  let A be set;
  thus A is preBoolean implies for X,Y being set st X in A & Y in A holds X \/
  Y in A & X \ Y in A by Def1,Def3;
  assume
A1: for X,Y being set st X in A & Y in A holds X \/ Y in A & X \ Y in A;
A2: A is diff-closed
  by A1;
  A is cup-closed
  by A1;
  hence thesis by A2;
end;
