
theorem Th29:
  for T being non empty TopSpace, B being prebasis of T holds
  B \/ {the carrier of T} is prebasis of T
proof
  let T be non empty TopSpace, B be prebasis of T;
  set C = B \/ {the carrier of T};
A1: the carrier of T in the topology of T by PRE_TOPC:def 1;
A2: B c= the topology of T by TOPS_2:64;
A3: {the carrier of T} c= the topology of T by A1,ZFMISC_1:31;
  then C c= the topology of T by A2,XBOOLE_1:8;
  then reconsider C as Subset-Family of T by XBOOLE_1:1;
A4: C c= the topology of T by A2,A3,XBOOLE_1:8;
A5: FinMeetCl B c= FinMeetCl C by CANTOR_1:14,XBOOLE_1:7;
  FinMeetCl B is Basis of T by Th23;
  hence thesis by A4,A5,CANTOR_1:def 4,TOPS_2:64;
end;
