
theorem Th19:
  for T,S being TopSpace, R being Refinement of T,S for W being
Subset of R st W in the topology of T or W in the topology of S holds W is open
proof
  let T,S be TopSpace, R be Refinement of T,S;
  let W be Subset of R;
  (the topology of T) \/ the topology of S is prebasis of R by YELLOW_9:def 6;
  then
A1: (the topology of T) \/ the topology of S c= the topology of R by TOPS_2:64;
  assume W in the topology of T or W in the topology of S;
  then W in (the topology of T) \/ the topology of S by XBOOLE_0:def 3;
  hence W in the topology of R by A1;
end;
