reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;
reserve S,S1 for Subset-Family of X;
reserve FS for non empty Subset of Filters(X);
reserve X for infinite set;
reserve Y,Y1,Y2,Z for Subset of X;
reserve F,Uf for Filter of X;

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;
