
theorem
for X being set ex Y being set st card X = card Y & X /\ Y = {}
proof
let X be set;
consider Y being set such that
A: card X c= card Y & X /\ Y = {} by thre;
consider Z being set such that B: Z c= Y & card Z = card X by A,thre1;
take Z;
thus card X = card Z by B;
now assume C: X /\ Z <> {};
  set o = the Element of X /\ Z;
  o in X & o in Z by C,XBOOLE_0:def 4;
  hence contradiction by A,B,XBOOLE_0:def 4;
  end;
hence thesis;
end;
