
theorem
  for T1, T2 being TopStruct st the carrier of T1 = the carrier of T2
  for T being Refinement of T1, T2
  holds T is TopExtension of T1 & T is TopExtension of T2
proof
  let T1, T2 be TopStruct such that
A1: the carrier of T1 = the carrier of T2;
  let T be Refinement of T1, T2;
A2: the carrier of T = (the carrier of T1) \/ the carrier of T2 by Def6
    .= the carrier of T1 by A1;
A3: (the topology of T1) \/ the topology of T2 is prebasis of T by Def6;
A4: the topology of T1 c= (the topology of T1) \/ the topology of T2 by
XBOOLE_1:7;
A5: the topology of T2 c= (the topology of T1) \/ the topology of T2 by
XBOOLE_1:7;
A6: (the topology of T1) \/ the topology of T2 c= the topology of T by A3,
TOPS_2:64;
  then
A7: the topology of T1 c= the topology of T by A4;
  the topology of T2 c= the topology of T by A5,A6;
  hence thesis by A1,A2,A7,Def5;
end;
