
theorem :: Theorem 1.3.2. (f)
  for T being TopSpace, A being Subset of T holds the carrier of T = Int
  A \/ Fr A \/ Int A`
proof
  let T be TopSpace, A be Subset of T;
  Int A \/ Fr A \/ Int A` = Int A \/ Int A` \/ Fr A by XBOOLE_1:4
    .= Int A \/ Int A` \/ (Cl A /\ Cl A`) by TOPS_1:def 2
    .= (Int A \/ Int A` \/ Cl A) /\ (Int A \/ Int A` \/ Cl A`) by XBOOLE_1:24
    .= (Int A \/ (Cl A)` \/ Cl A) /\ (Int A \/ Int A` \/ Cl A`) by TDLAT_3:3
    .= (Int A \/ ((Cl A)` \/ Cl A)) /\ (Int A \/ Int A` \/ Cl A`) by XBOOLE_1:4
    .= (Int A \/ [#]T) /\ (Int A \/ Int A` \/ Cl A`) by PRE_TOPC:2
    .= (Int A \/ [#]T) /\ (Int A \/ Int A` \/ (Int A)`) by TDLAT_3:2
    .= (Int A \/ [#]T) /\ (Int A \/ (Int A)` \/ Int A`) by XBOOLE_1:4
    .= (Int A \/ [#]T) /\ ([#]T \/ Int A`) by PRE_TOPC:2
    .= [#]T /\ ([#]T \/ Int A`) by XBOOLE_1:12
    .= [#]T /\ [#]T by XBOOLE_1:12
    .= [#]T;
  hence thesis;
end;
