reserve X,Y,x for set;
reserve A for non empty preBoolean set;

theorem Th8:
  for A being set holds bool A is preBoolean
proof
  let A be set;
  now
    let X,Y be set;
    assume X in bool A & Y in bool A;
    then reconsider X9=X,Y9=Y as Subset of A;
    X9 \/ Y9 in bool A & X9 \ Y9 in bool A;
    hence X \/ Y in bool A & X \ Y in bool A;
  end;
  hence thesis by Th1;
end;
