
theorem Th4:
  for T being TopSpace, A,B being Subset of T st A is open & B is
  closed holds A \ B is open
proof
  let T be TopSpace, A,B be Subset of T;
  assume that
A1: A is open and
A2: B is closed;
  [#](T)\B is open by A2,PRE_TOPC:def 3;
  then A /\ ([#](T)\B) is open by A1,TOPS_1:11;
  then
A3: A /\ [#](T) \ A /\ B is open by XBOOLE_1:50;
  A /\ [#](T) = A by XBOOLE_1:28;
  hence thesis by A3,XBOOLE_1:47;
end;
