reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;

theorem
  A \/ B = the carrier of X implies (A is closed implies A \/ Int B =
  the carrier of X)
proof
  assume A \/ B = the carrier of X;
  then (A \/ B)` = {}X by XBOOLE_1:37;
  then ( A`) /\ B` = {} by XBOOLE_1:53;
  then
A1: ( A`) misses B`;
  assume A is closed;
  then ( A`) misses Cl B` by A1,TSEP_1:36;
  then ( A`) /\ (Cl B`)`` = {};
  then ( A`) /\ ((Int B)`) = {} by TOPS_1:def 1;
  then (A \/ Int B)` = {}X by XBOOLE_1:53;
  then (A \/ Int B)`` = [#]X;
  hence thesis;
end;
