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

theorem Th21: :: Theorem 6
  Border A is regular_open & Border A = (Int Cl A) \ (Cl Int A) &
  Border A = Int Cl A /\ (Int Cl A`)
proof
  Fr A = Cl Fr A by PRE_TOPC:22;
  hence Border A is regular_open;
  (Int Cl A) \ (Cl Int A) = (Int Cl A) \ ((Cl Int A)`)`
    .= (Int Cl A) \ (Int (Int A)`)` by TDLAT_3:3
    .= (Int Cl A) \ (Int Cl A`)` by TDLAT_3:2
    .= Int Cl A /\ (Int Cl A`)`` by SUBSET_1:13
    .= Int (Cl A /\ Cl A`) by TOPS_1:17
    .= Int Fr A by TOPS_1:def 2;
  hence Border A = (Int Cl A) \ (Cl Int A);
  Int Cl A /\ (Int Cl A`) = Int (Cl A /\ Cl A`) by TOPS_1:17
    .= Int Fr A by TOPS_1:def 2;
  hence thesis;
end;
