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

theorem For191: :: (19)
  (delta_1 R).(X,Y) = card (X \+\ Y) / card (X \/ Y)
  proof
    per cases;
    suppose
A0:   X \/ Y <> {}; then
A2:   (CMap kappa_1 R).(Y,X) = card (Y \ X) / card (X \/ Y) by PropEx3;
      (delta_1 R).(X,Y) = (CMap kappa_1 R).(X,Y) + (CMap kappa_1 R).(Y,X)
        by Delta1
      .= card (X \ Y) / card (X \/ Y) + card (Y \ X) / card (X \/ Y)
        by A0,PropEx3,A2
      .= (card (X \ Y) + card (Y \ X)) / card (X \/ Y) by XCMPLX_1:62
      .= card (X \+\ Y) / card (X \/ Y) by XBOOLE_1:82,CARD_2:40;
       hence thesis;
    end;
    suppose
A0:   X \/ Y = {}; then
      X = {} & Y = {}; then
      (CMap kappa_1 R).(X,Y) = 0 & (CMap kappa_1 R).(Y,X) = 0 by Prop6a; then
      (delta_1 R).(X,Y) = 0 + 0 by Delta1;
      hence thesis by A0;
    end;
  end;
