
theorem Th23:

:: YELLOW_9:58 revised
  for T1,T2 being non empty TopSpace st the carrier of T1 = the
  carrier of T2 for T be Refinement of T1, T2 for B1 being prebasis of T1, B2
  being prebasis of T2 holds B1 \/ B2 is prebasis of T
proof
  let T1,T2 be non empty TopSpace such that
A1: the carrier of T1 = the carrier of T2;
  let T be Refinement of T1, T2;
  let B1 be prebasis of T1, B2 be prebasis of T2;
  the carrier of T = (the carrier of T1) \/ the carrier of T2 by YELLOW_9:def 6
    .= the carrier of T1 by A1;
  then {the carrier of T1, the carrier of T2} = {the carrier of T} by A1,
ENUMSET1:29;
  then reconsider K = B1 \/ B2 \/ {the carrier of T} as prebasis of T by
YELLOW_9:58;
  B1 \/ B2 c= K by XBOOLE_1:7;
  then reconsider K9 = B1 \/ B2 as Subset-Family of T by XBOOLE_1:1;
  K c= FinMeetCl K9
  proof
    let a be object;
    assume a in K;
    then
    a in K9 & K9 c= FinMeetCl K9 or a in {the carrier of T} & the carrier
    of T in FinMeetCl K9 by CANTOR_1:4,8,XBOOLE_0:def 3;
    hence thesis by TARSKI:def 1;
  end;
  then FinMeetCl K c= FinMeetCl FinMeetCl K9 by CANTOR_1:14;
  then
A2: FinMeetCl K c= FinMeetCl K9 by CANTOR_1:11;
  FinMeetCl K9 c= FinMeetCl K by CANTOR_1:14,XBOOLE_1:7;
  then FinMeetCl K9 = FinMeetCl K by A2;
  then FinMeetCl K9 is Basis of T by YELLOW_9:23;
  hence thesis by YELLOW_9:23;
end;
