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 Th20:
  Frechet_Filter(X) c= F implies F is uniform
proof
  assume
A1: Frechet_Filter(X) c= F;
  let Y;
  assume Y in F;
  then not X \ Y in Frechet_Filter(X) by A1,Th6;
  then
A2: not card (X \ (X \ Y)) in card X;
A3: card Y c= card X by CARD_1:11;
  X \ (X \ Y) = X /\ Y by XBOOLE_1:48
    .= Y by XBOOLE_1:28;
  then card X c= card Y by A2,CARD_1:4;
  hence thesis by A3;
end;
