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

theorem PropEx31:
  (CMap kappa_2 R).(X,Y) = card (X \ Y) / card [#]R
  proof
    (X` \/ Y)` = (X \ Y)`` by SUBSET_1:14; then
A3: card [#]R - card (X` \/ Y) = card (X \ Y) by CARD_2:44;
    (CMap kappa_2 R).(X,Y) = 1 - (kappa_2 R).(X,Y) by CDef
      .= 1 - kappa_2(X,Y) by ROUGHIF1:def 6
      .= (card [#]R / card [#]R) - card (X` \/ Y) / card [#]R
       by XCMPLX_1:60
      .= card (X \ Y) / card ([#]R) by A3,XCMPLX_1:120;
    hence thesis;
  end;
