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

theorem :: Proposition 7 a2)
  (delta_2 R).(X,Y) = 0 iff X = Y
  proof
B1: (CMap kappa_2 R).(X,Y) >= 0 & (CMap kappa_2 R).(Y,X) >= 0 by XXREAL_1:1;
    hereby assume (delta_2 R).(X,Y) = 0; then
      (CMap kappa_2 R).(X,Y) + (CMap kappa_2 R).(Y,X) = 0 by Delta2; then
      (CMap kappa_2 R).(X,Y) = 0 & (CMap kappa_2 R).(Y,X) = 0 by B1;
      hence X = Y by Prop6a;
    end;
    assume
A1: X = Y;
    (delta_2 R).(X,Y) = (CMap kappa_2 R).(X,Y) + (CMap kappa_2 R).(Y,X)
      by Delta2
       .= (CMap kappa_2 R).(X,X) + 0 by Prop6a,A1
       .= 0 + 0 by Prop6a;
    hence thesis;
  end;
