
theorem Lemacik:
  for A, B be finite set st A \/ B <> {} holds
    1 - card (A /\ B) / card (A \/ B) = card (A \+\ B) / card (A \/ B)
  proof
    let A,B be finite set;
    assume
A1: A \/ B <> {};
B1: A /\ B c= A by XBOOLE_1:17;
    A c= A \/ B by XBOOLE_1:7; then
B2: A /\ B c= A \/ B by B1;
    A \+\ B = (A \/ B) \ A /\ B by XBOOLE_1:101; then
B3: card (A \+\ B) = card (A \/ B) - card (A /\ B) by CARD_2:44,B2;
    1 - card (A /\ B) / card (A \/ B) =
      card (A \/ B) / card (A \/ B) - card (A /\ B) / card (A \/ B)
        by A1,XCMPLX_1:60
       .= card (A \+\ B) / card (A \/ B) by B3,XCMPLX_1:120;
    hence thesis;
  end;
