
theorem Th25:
  for T1, T2 being TopSpace, B being Basis of T1
  st the carrier of T1 = the carrier of T2 & B is Basis of T2
  holds the topology of T1 = the topology of T2
proof
  let T1, T2 be TopSpace, B be Basis of T1;
  assume that
A1: the carrier of T1 = the carrier of T2 and
A2: B is Basis of T2;
  reconsider C = B as Basis of T2 by A2;
  thus the topology of T1 = UniCl C by A1,Th22
    .= the topology of T2 by Th22;
end;
