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 Th19:
  Y in Frechet_Ideal(X) iff card Y in card X
proof
  Y in Frechet_Ideal(X) iff Y` in Frechet_Filter(X) by SETFAM_1:def 7;
  then Y in Frechet_Ideal(X) iff card Y`` in card X by Th18;
  hence thesis;
end;
