for X1, Y1, X being finite set st ( Y1 is empty implies X1 is empty ) & X c= X1 holds
for F being Function of X1,Y1 st F is one-to-one & card X1 = card Y1 holds
((card X1) -' (card X)) ! = card { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) }