
theorem Th3:
  for T being non empty TopSpace, x being Point of T, B being Basis
  of x holds B <> {}
proof
  let T be non empty TopSpace, x be Point of T, B be Basis of x;
A1: the carrier of T in the topology of T by PRE_TOPC:def 1;
  set U1=[#]T;
  reconsider T as non empty TopStruct;
  U1 is open by A1,PRE_TOPC:def 2;
  then ex U2 be Subset of T st U2 in B & U2 c= U1 by YELLOW_8:13;
  hence thesis;
end;
