 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 :: false for X = {}!
  X <> {} implies 1 - kappa (X,Y) = kappa (X,Y`)
  proof
    assume
A1: X <> {};
a2: Y \/ ([#]R \ Y) = [#]R by XBOOLE_1:45;
    kappa (X,Y \/ Y`) = kappa (X,Y) + kappa (X,Y`)
      by A1,Prop1e,SUBSET_1:23; then
    kappa (X,Y) + kappa (X,Y`) = 1 by a2,Prop1a;
    hence thesis;
  end;
