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

theorem
  for A being non empty set st for X,Y being Element of A holds X \+\ Y
  in A & X /\ Y in A holds A is preBoolean
proof
  let A be non empty set such that
A1: for X,Y being Element of A holds X \+\ Y in A & X /\ Y in A;
  now
    let X,Y be set;
    assume that
A2: X in A and
A3: Y in A;
    reconsider Z1 = X \+\ Y, Z2 = X /\ Y as Element of A by A1,A2,A3;
    X \/ Y = Z1 \+\ Z2 by XBOOLE_1:94;
    hence X \/ Y in A by A1;
    X \ Y = X \+\ Z2 by XBOOLE_1:100;
    hence X \ Y in A by A1,A2;
  end;
  hence thesis by Th1;
end;
