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

theorem Th22: :: Theorem 7
  A is supercondensed iff Int A is regular_open & Border A is empty
proof
A1: Int A c= Int Cl A by PRE_TOPC:18,TOPS_1:19;
  thus A is supercondensed implies Int A is regular_open & Border A is empty
  proof
    assume
A2: A is supercondensed;
    then Int Cl A = Int A;
    then Int Cl A c= Cl Int A by PRE_TOPC:18;
    then (Int Cl A) \ (Cl Int A) is empty by XBOOLE_1:37;
    hence thesis by A2,Th21;
  end;
  assume that
A3: Int A is regular_open 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: Int Int Cl A c= Int Cl Int A by TOPS_1:19;
  Int A = Int Cl Int A by A3,TOPS_1:def 8;
  then Int Cl A = Int A by A5,A1,XBOOLE_0:def 10;
  hence thesis;
end;
