reserve a,b,c for set;

theorem
  for T being non empty TopSpace holds T is first-countable iff Chi(T) c= omega
proof
  let T be non empty TopSpace;
  set X = the set of all Chi(x,T) where x is Point of T;
A1: Chi(T) = union X by Th4;
  thus T is first-countable implies Chi(T) c= omega
  proof
    assume
A2: for x be Point of T ex B be Basis of x st B is countable;
    now
      let a;
      assume a in X;
      then consider x being Point of T such that
A3:   a = Chi(x,T);
      consider B being Basis of x such that
A4:   B is countable by A2;
A5:   card B c= omega by A4;
      Chi(x,T) c= card B by Def1;
      hence a c= omega by A3,A5;
    end;
    hence thesis by A1,ZFMISC_1:76;
  end;
  assume
A6: Chi(T) c= omega;
  let x be Point of T;
  consider B being Basis of x such that
A7: Chi(x,T) = card B by Def1;
  take B;
  Chi(x,T) c= Chi(T) by Th5;
  hence card B c= omega by A6,A7;
end;
