
theorem LemmaCard:
  for R being finite set
  for X,Y being Subset of R holds
  card (X \/ Y) = card (X /\ Y) iff X = Y
  proof
    let R be finite set;
    let X,Y be Subset of R;
    hereby
      assume card (X \/ Y) = card (X /\ Y); then
A2:   X /\ Y = X \/ Y by CARD_2:102,XBOOLE_1:29; then
      X = X \/ (X \/ Y) by XBOOLE_1:22; then
      X = X \/ Y by XBOOLE_1:6;
      hence X = Y by XBOOLE_1:21,A2;
    end;
    assume X = Y;
    hence thesis;
  end;
