
theorem
  for T1,T2 being TopStruct, T being Refinement of T1,T2
  for X being Subset-Family of T
  st X = (the topology of T1) \/ (the topology of T2)
  holds the topology of T = UniCl FinMeetCl X
proof
  let T1,T2 be TopStruct, T be Refinement of T1,T2;
  let X be Subset-Family of T;
  assume X = (the topology of T1) \/ (the topology of T2);
  then X is prebasis of T by Def6;
  then FinMeetCl X is Basis of T by Th23;
  hence thesis by Th22;
end;
