reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;
reserve d for non zero Nat;
reserve i,i0,i1 for Element of Seg d;

theorem Th7:
  for X,Y being finite set holds
  (card X is even iff card Y is even) iff card(X \+\ Y) is even
proof
  let X,Y be finite set;
A1: X \ Y misses X /\ Y by XBOOLE_1:89;
A2: X = (X \ Y) \/ (X /\ Y) by XBOOLE_1:51;
A3: Y \ X misses X /\ Y by XBOOLE_1:89;
A4: Y = (Y \ X) \/ (X /\ Y) by XBOOLE_1:51;
A5: X \ Y misses Y \ X by XBOOLE_1:82;
A6: X \+\ Y = (X \ Y) \/ (Y \ X) by XBOOLE_0:def 6;
A7: (card(X \ Y) is even iff card(X /\ Y) is even) iff card X is even by A1,A2
,Th6;
  (card(Y \ X) is even iff card(X /\ Y) is even) iff card Y is even by A3,A4
,Th6;
  hence thesis by A5,A6,A7,Th6;
end;
