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

theorem Prop6e1: :: Proposition 6 e1)
  0 <= (CMap kappa_1 R).(X,Y) + (CMap kappa_1 R).(Y,X) <= 1
  proof
    per cases;
    suppose X \/ Y = {}; then
      (CMap kappa_1 R).(X,Y) = 0 & (CMap kappa_1 R).(Y,X) = 0 by PropEx30;
      hence thesis;
    end;
    suppose
B1:    X \/ Y <> {}; then
A1:   (CMap kappa_1 R).(X,Y) + (CMap kappa_1 R).(Y,X) =
        card (X \ Y) / card (X \/ Y) + (CMap kappa_1 R).(Y,X) by PropEx3
        .= card (X \ Y) / card (X \/ Y) + card (Y \ X) / card (X \/ Y)
          by PropEx3,B1
        .= (card (X \ Y) + card (Y \ X)) / card (X \/ Y) by XCMPLX_1:62
        .= card (X \+\ Y) / card (X \/ Y) by CARD_2:40,XBOOLE_1:82;
      card (X \+\ Y) <= card (X \/ Y) by Lack,NAT_1:43;
      hence thesis by A1,XREAL_1:183;
    end;
  end;
