 reserve R for 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite Approximation_Space;
 reserve X,Y,Z,W for Subset of R;

theorem :: Proposition 2 d)
  X <> {} & X misses Y implies
    kappa (X, Z \ Y) = kappa (X, Z \/ Y) = kappa (X,Z)
  proof
    assume
A1: X <> {} & X misses Y; then
D1: kappa (X,Y) = 0 by Prop2b;
D3: kappa (X,Z) = kappa (X, (Z /\ Y) \/ (Z \ Y)) by XBOOLE_1:51
               .= kappa (X, Z /\ Y) + kappa (X, Z \ Y)
                   by Prop1e,A1,XBOOLE_1:89;
    kappa (X,Z /\ Y) <= 0 by D1,Prop1b,XBOOLE_1:17; then
D5: kappa (X,Z /\ Y) = 0 by XXREAL_1:1;
e1: kappa (X, Z \/ Y) <= kappa (X,Z) + kappa (X,Y) by Prop1d;
    kappa (X,Z) <= kappa (X, Z \/ Y) by Prop1b,XBOOLE_1:7;
    hence thesis by D5,D3,e1,D1,XXREAL_0:1;
  end;
