 reserve R for finite Approximation_Space;
 reserve X,Y,Z for Subset of R;

theorem PropEx3k:
  X <> {} implies (CMap kappa R).(X,Y) = card (X \ Y) / card X
  proof
    assume
A1: X <> {};
    X \ Y = X \ (X /\ Y) by XBOOLE_1:47; then
T1: card (X \ Y) = card X - card (X /\ Y) by XBOOLE_1:17,CARD_2:44;
    (CMap kappa R).(X,Y) = 1 - (kappa R).(X,Y) by CDef
      .= 1 - kappa (X,Y) by ROUGHIF1:def 2
      .= 1 - card (X /\ Y) / card X by A1,ROUGHIF1:def 1
      .= (card X / card X) - card (X /\ Y) / card X by A1,XCMPLX_1:60
      .= card (X \ Y) / card X by T1,XCMPLX_1:120;
    hence thesis;
  end;
