
theorem HoHo:
  for x1,x2 being finite set holds
    (2 * card (x1 \+\ x2)) / (card x1 + card x2 + card (x1 \+\ x2))
       = card (x1 \+\ x2) / card (x1 \/ x2)
  proof
    let x1,x2 be finite set;
    set p = card (x1 \/ x2);
    set q = card x1 + card x2 + card (x1 \+\ x2);
    set r = card (x1 \+\ x2);
    x1 \/ x2 = x1 \/ (x2 \ x1) by XBOOLE_1:39; then
HH: p = card x1 + card (x2 \ x1) by XBOOLE_1:79,CARD_2:40;
    x1 = (x1 \ x2) \/ (x1 /\ x2) by XBOOLE_1:51; then
g1: card x1 = card (x1 \ x2) + card (x1 /\ x2) by CARD_2:40,XBOOLE_1:89;
    x2 = (x2 \ x1) \/ (x1 /\ x2) by XBOOLE_1:51; then
g2: card x2 = card (x2 \ x1) + card (x1 /\ x2) by CARD_2:40,XBOOLE_1:89;
    q = (card (x1 \ x2) + card (x1 /\ x2)) +
       (card (x2 \ x1) + card (x1 /\ x2)) +
         (card (x1 \ x2) + card (x2 \ x1)) by CARD_2:40,XBOOLE_1:82,g1,g2
      .= 2 * p by g1,HH;
    hence thesis by XCMPLX_1:91;
  end;
