theorem
  Int(K \ L) c= Int K \ Int L
proof
A1: Int(K \ L) = (Cl (K /\ L`)`)` by SUBSET_1:13
    .= (Cl((K`) \/ L``))` by XBOOLE_1:54
    .= ((Cl K` \/ Cl L))` by PRE_TOPC:20
    .= ((Cl L)`) /\ Int K by XBOOLE_1:53;
  L c= Cl L by PRE_TOPC:18;
  then
A2: (Cl L)` c= L` by SUBSET_1:12;
  Int L c= L by Th16;
  then L` c= (Int L)` by SUBSET_1:12;
  then ((Cl L)`) c= (Int L)` by A2;
  then Int(K \ L) c= Int K /\ (Int L)` by A1,XBOOLE_1:26;
  hence thesis by SUBSET_1:13;
end;
