reserve a,b,c for set;

theorem Th17:
  for T being TopSpace for B0 being Basis of T for f being
  Function of the carrier of T, the topology of T st B0 = rng f & for x being
Point of T holds x in f.x & for U being open Subset of T st x in U holds f.x c=
  U holds weight T = card B0
proof
  let T be TopSpace;
  let B0 be Basis of T;
  let f be Function of the carrier of T, the topology of T such that
A1: B0 = rng f and
A2: for x being Point of T holds x in f.x & for U being open Subset of T
  st x in U holds f.x c= U;
  set M = the set of all card C where C is Basis of T;
A3: card B0 in M;
  weight T = meet M by WAYBEL23:def 5;
  hence weight T c= card B0 by A3,SETFAM_1:3;
  ex B being Basis of T st card B = weight T by WAYBEL23:74;
  hence thesis by A1,A2,Th16,CARD_1:11;
end;
