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

theorem Prop6d2: :: Proposition 6 d2)
  (CMap kappa_2 R).(X,Y) + (CMap kappa_2 R).(Y,Z) >= (CMap kappa_2 R).(X,Z)
  proof
B1: (CMap kappa_2 R).(X,Y) = card (X \ Y) / card [#]R by PropEx31;
B2: (CMap kappa_2 R).(Y,Z) = card (Y \ Z) / card [#]R by PropEx31;
B3: (CMap kappa_2 R).(X,Z) = card (X \ Z) / card [#]R by PropEx31;
A1: Y \ Z c= Y by XBOOLE_1:36;
    X \ Y misses Y by XBOOLE_1:79; then
A2: card ((X \ Y) \/ (Y \ Z)) = card (X \ Y) + card (Y \ Z)
      by CARD_2:40,A1,XBOOLE_1:63;
    X \ Z c= (X \ Y) \/ (Y \ Z)
    proof
      let x be object;
      assume x in X \ Z; then
A3:   x in X & not x in Z by XBOOLE_0:def 5;
      per cases;
      suppose x in Y; then
        x in Y \ Z by A3,XBOOLE_0:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose not x in Y; then
        x in X \ Y by A3,XBOOLE_0:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
    end; then
    (card (X \ Y) + card (Y \ Z)) / card [#]R >= card (X \ Z) / card [#]R
       by A2,XREAL_1:72,NAT_1:43;
    hence thesis by B1,B2,B3,XCMPLX_1:62;
  end;
