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

theorem :: Proposition 6 a), specified for kappa, unnecessary
  (CMap kappa R).(X,Y) = 0 iff X c= Y
  proof
    thus (CMap kappa R).(X,Y) = 0 implies X c= Y
    proof
      assume (CMap kappa R).(X,Y) = 0; then
      1 - (kappa R).(X,Y) = 0 by CDef; then
      kappa (X,Y) = 1 by ROUGHIF1:def 2;
      hence thesis by ROUGHIF1:6;
    end;
    assume X c= Y; then
A1: kappa (X,Y) = 1 by ROUGHIF1:6;
    (CMap kappa R).(X,Y) = 1 - (kappa R).(X,Y) by CDef
        .= 1 - 1 by A1,ROUGHIF1:def 2;
    hence thesis;
  end;
