theorem Th18:
  Y in Frechet_Filter(X) iff card (X \ Y) in card X
proof
  thus Y in Frechet_Filter(X) implies card (X \ Y) in card X
  proof
    defpred P[set] means card (X \ $1) in card X;
    assume Y in Frechet_Filter(X);
    then
A1: Y in {Y1: P[Y1]};
    thus P[Y] from ElemProp(A1);
  end;
  thus thesis;
end;
