reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th2:
  for a,b being set st card a = n + k & card b = n + k holds card(a
  /\ b) = n iff card(a \/ b) = n + 2*k
proof
  let a,b be set;
  assume that
A1: card a = n + k and
A2: card b = n + k;
A3: a is finite by A1;
A4: b is finite by A2;
  thus card(a /\ b) = n implies card(a \/ b) = n + 2*k
  proof
    assume card(a /\ b) = n;
    then card(a \/ b) = n + k + (n + k) - n by A1,A2,A3,A4,CARD_2:45;
    hence thesis;
  end;
  thus card (a \/ b) = n + 2*k implies card (a /\ b) = n
  proof
    reconsider m = card (a /\ b) as Nat by A3;
    assume card(a \/ b) = n + 2*k;
    then n + 2*k = n + k + (n + k) - m by A1,A2,A3,A4,CARD_2:45;
    hence thesis;
  end;
end;
