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

theorem Th4:
  Int(A \/ B) c= (Cl A) \/ Int B
proof
  (Int A`) /\ Cl B` c= Cl(( A`) /\ B`) by Th3;
  then (Cl(( A`) /\ B`))` c= ((Int A`) /\ Cl B`)` by SUBSET_1:12;
  then Int((( A`) /\ B`)`) c= (((Int A`) /\ Cl B`))` by TDLAT_3:3;
  then Int(( A``) \/ ( B``)) c= (((Int A`) /\ Cl B`))` by XBOOLE_1:54;
  then Int(A \/ B) c= (Int A`)` \/ (Cl B`)` by XBOOLE_1:54;
  then Int(A \/ B) c= Cl A \/ (Cl B`)` by TDLAT_3:1;
  hence thesis by TOPS_1:def 1;
end;
