
theorem Th26:
  for T1, T2 being TopSpace, P being prebasis of T1
  st the carrier of T1 = the carrier of T2 & P is prebasis of T2
  holds the topology of T1 = the topology of T2
proof
  let T1, T2 be TopSpace, P be prebasis of T1;
  assume that
A1: the carrier of T1 = the carrier of T2 and
A2: P is prebasis of T2;
  reconsider C = P as prebasis of T2 by A2;
A3: FinMeetCl P is Basis of T1 by Th23;
  FinMeetCl C is Basis of T2 by Th23;
  hence thesis by A1,A3,Th25;
end;
