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

theorem PropEx3:
  X \/ Y <> {} implies
    (CMap kappa_1 R).(X,Y) = card (X \ Y) / card (X \/ Y)
  proof
    assume
A0: X \/ Y <> {};
A1: (CMap kappa_1 R).(X,Y) = 1 - (kappa_1 R).(X,Y) by CDef
       .= 1 - kappa_1 (X,Y) by ROUGHIF1:def 5;
    X \ Y = (X \/ Y) \ Y by XBOOLE_1:40; then
X1: card (X \ Y) = card (X \/ Y) - card Y by CARD_2:44,XBOOLE_1:7;
    1 - card Y / card (X \/ Y) = (card (X \/ Y) / card (X \/ Y)) -
            card Y / card (X \/ Y) by A0,XCMPLX_1:60
       .= card (X \ Y) / card (X \/ Y) by X1,XCMPLX_1:120;
    hence thesis by A1,A0,ROUGHIF1:def 3;
  end;
