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

theorem :: Theorem 5
  A is condensed iff Fr A = Cl Int A \ Int Cl A & Fr A = Cl Int A /\ Cl Int A`
proof
A1: A c= Cl A by PRE_TOPC:18;
  Cl Int A /\ Cl Int A` c= Cl Int A by XBOOLE_1:17;
  then
A2: Int A \/ (Cl Int A /\ Cl Int A`) c= Int A \/ Cl Int A by XBOOLE_1:13;
  thus A is condensed implies Fr A = Cl Int A \ Int Cl A & Fr A = Cl Int A /\
  Cl Int A`
  proof
    assume
A3: A is condensed;
    then A` is condensed by TDLAT_1:16;
    then
A4: Cl Int A` = Cl A`;
    thus thesis by A3,A4,TOPS_1:def 2;
  end;
  assume that
  Fr A = Cl Int A \ Int Cl A and
A5: Fr A = Cl Int A /\ Cl Int A`;
A6: Cl A \/ Int A = Int A \/ Fr A by XBOOLE_1:39;
  Int A c= A by TOPS_1:16;
  then Cl A = Int A \/ (Cl Int A /\ Cl Int A`) by A5,A1,A6,XBOOLE_1:1,12;
  then
A7: Cl A c= Cl Int A by A2,PRE_TOPC:18,XBOOLE_1:12;
  Cl Int A c= Cl A by PRE_TOPC:19,TOPS_1:16;
  then Cl Int A = Cl A by A7,XBOOLE_0:def 10;
  then
A8: A is subcondensed;
A9: A` c= Cl A` by PRE_TOPC:18;
A10: Cl A` \/ Int A` = Int A` \/ Fr A` by XBOOLE_1:39;
A11: Fr A = Fr A` by TOPS_1:29;
  Cl Int A` /\ Cl Int A`` c= Cl Int A` by XBOOLE_1:17;
  then
A12: Int A` \/ (Cl Int A` /\ Cl Int A``) c= Int A` \/ Cl Int A` by XBOOLE_1:13;
A13: Cl Int A` c= Cl A` by PRE_TOPC:19,TOPS_1:16;
A14: Cl Int A` = Cl (Cl A)` by TDLAT_3:3
    .= (Int Cl A)` by TDLAT_3:2;
A15: Int A` \/ Cl Int A` = Cl Int A` by PRE_TOPC:18,XBOOLE_1:12;
  Int A` c= A` by TOPS_1:16;
  then Cl A` = Int A` \/ (Cl Int A` /\ Cl Int A``) by A5,A9,A11,A10,XBOOLE_1:1
,12;
  then
A16: Cl A` = Cl Int A` by A13,A12,A15,XBOOLE_0:def 10;
  Cl A` = (Int A)` by TDLAT_3:2;
  then Int A = (Int Cl A)`` by A16,A14;
  then A is supercondensed;
  hence thesis by A8;
end;
