 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 Prop4b: :: Proposition 4 b)
  kappa_2(X,Y) = 0 iff X = [#]R & Y = {}
  proof
    thus kappa_2(X,Y) = 0 implies X = [#]R & Y = {}
    proof
      assume
ac:   kappa_2(X,Y) = 0; then
      X` = {} & Y = {}; then
      X`` = ({}R)`;
      hence thesis by ac;
    end;
    assume
A2: X = [#]R & Y = {}; then
    X` = ({}R)``;
    hence thesis by A2;
  end;
