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

theorem Th23: :: Theorem 7
  A is subcondensed iff Cl A is regular_closed & Border A is empty
proof
A1: Cl Int A c= Cl A by PRE_TOPC:19,TOPS_1:16;
  thus A is subcondensed implies Cl A is regular_closed & Border A is empty
  proof
    assume
A2: A is subcondensed;
    then Cl Int A = Cl A;
    then Int Cl A c= Cl Int A by TOPS_1:16;
    then (Int Cl A) \ (Cl Int A) is empty by XBOOLE_1:37;
    hence thesis by A2,Th21;
  end;
  assume that
A3: Cl A is regular_closed and
A4: Border A is empty;
  (Int Cl A) \ (Cl Int A) is empty by A4,Th21;
  then Int Cl A c= Cl Int A by XBOOLE_1:37;
  then
A5: Cl Int Cl A c= Cl Cl Int A by PRE_TOPC:19;
  Cl A = Cl Int Cl A by A3,TOPS_1:def 7;
  then Cl Int A = Cl A by A5,A1,XBOOLE_0:def 10;
  hence thesis;
end;
