 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 Prop4a: :: Proposition 4 a)
  X <> {} implies (kappa_1 (X,Y) = 0 iff Y = {})
  proof
    assume
a3: X <> {};
    thus kappa_1 (X,Y) = 0 implies Y = {} by LemmaProp4a;
    assume
A2: Y = {};
    kappa_1 (X,Y) = card Y / card (X \/ Y) by Kappa1,a3
       .= 0 / card (X \/ Y) by A2;
    hence thesis;
  end;
