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
  Uf is uniform being_ultrafilter implies Frechet_Filter(X) c= Uf
proof
  assume
A1: Uf is uniform being_ultrafilter;
  thus Frechet_Filter(X) c= Uf
  proof
    let Y be object;
     reconsider YY=Y as set by TARSKI:1;
    assume
A2: Y in Frechet_Filter(X);
    then card (X \ YY) in card X by Th18;
    then card (X \ YY) <> card X;
    then not (X \ YY) in Uf by A1;
    hence thesis by A1,A2;
  end;
end;
