reserve x for set;

theorem Th22:
  for T1,T2 being TopSpace, T being non empty TopSpace st T is
  TopExtension of T1 & T is TopExtension of T2 for R being Refinement of T1,T2
  holds T is TopExtension of R
proof
  let T1,T2 be TopSpace, T be non empty TopSpace such that
A1: the carrier of T1 = the carrier of T and
A2: the topology of T1 c= the topology of T and
A3: the carrier of T2 = the carrier of T and
A4: the topology of T2 c= the topology of T;
  let R be Refinement of T1, T2;
A5: the carrier of R = (the carrier of T) \/ (the carrier of T) by A1,A3,
YELLOW_9:def 6;
  hence the carrier of R = the carrier of T;
  reconsider S = (the topology of T1) \/ (the topology of T2) as prebasis of R
  by YELLOW_9:def 6;
  FinMeetCl S is Basis of R by YELLOW_9:23;
  then
A6: UniCl FinMeetCl S = the topology of R by YELLOW_9:22;
  S c= the topology of T by A2,A4,XBOOLE_1:8;
  then FinMeetCl S c= FinMeetCl the topology of T by A5,CANTOR_1:14;
  then the topology of R c= UniCl FinMeetCl the topology of T by A5,A6,
CANTOR_1:9;
  hence thesis by CANTOR_1:7;
end;
