reserve a,b,c for set;
reserve r for Real,
  X for set,
  n for Element of NAT;

theorem
  for X being infinite set for U,V being non empty open Subset of
  ClFinTop(X) holds U meets V
proof
  let X be infinite set;
  let U,V be non empty open Subset of ClFinTop(X);
  assume U misses V;
  then
A1: U c= V` by SUBSET_1:23;
A2: the carrier of ClFinTop X = X by Def6;
  V` is finite by Th34;
  then U` is infinite by A1,A2,Th35;
  then U` = [#]the carrier of ClFinTop X by A2,Def6;
  then U`` = {}the carrier of ClFinTop X by XBOOLE_1:37;
  hence thesis;
end;
