reserve T for TopSpace;
reserve A,B for Subset of T;

theorem Th2:
  for C being Subset of T st C is open & A /\ C c= B holds C c= Int (A` \/ B)
proof
  let C be Subset of T;
  assume that
A1: C is open and
A2: A /\ C c= B;
A3: C c= C \/ A` by XBOOLE_1:7;
  (A /\ C) \/ A` = (A \/ A`) /\ (C \/ A`) by XBOOLE_1:24
    .= [#] T /\ (C \/ A`) by PRE_TOPC:2
    .= C \/ A` by XBOOLE_1:28;
  then C \/ A` c= B \/ A` by A2,XBOOLE_1:9;
  then C c= B \/ A` by A3;
  then Int(C) c= Int(A` \/ B) by TOPS_1:19;
  hence thesis by A1,TOPS_1:23;
end;
