reserve a,b,c for set;

theorem Th16:
  for T being TopSpace for B0,B 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 B0 c= B
proof
  let T be TopSpace;
  let B0,B be Basis of T;
  let f be Function of the carrier of T, the topology of T;
  assume
A1: B0 = rng f;
  assume
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;
  let a be object;
  assume
A3: a in B0;
  then reconsider V = a as Subset of T;
  consider b being object such that
A4: b in dom f and
A5: a = f.b by A1,A3,FUNCT_1:def 3;
A6: V is open by A3,YELLOW_8:10;
  reconsider b as Element of T by A4;
  b in V by A2,A5;
  then consider U being Subset of T such that
A7: U in B and
A8: b in U and
A9: U c= V by A6,YELLOW_9:31;
  U is open by A7,YELLOW_8:10;
  then f.b c= U by A2,A8;
  hence thesis by A7,A9,XBOOLE_0:def 10,A5;
end;
