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

theorem :: Corollary to Theorem 3:: Theorem 4
  A is condensed iff ex B st B is regular_open & B c= A & A c= Cl B
proof
  thus A is condensed implies ex B st B is regular_open & B c= A & A c= Cl B
  proof
    assume
A1: A is condensed;
    then
A2: Cl Int A = Cl A;
    take Int Cl A;
    Int Cl A = Int A by A1;
    hence thesis by A2,PRE_TOPC:18,TOPS_1:16;
  end;
  given B such that
A3: B is regular_open and
A4: B c= A and
A5: A c= Cl B;
A6: Int Cl B = B by A3,TOPS_1:def 8;
  Int B c= Int A by A4,TOPS_1:19;
  then
A7: Cl Int B c= Cl Int A by PRE_TOPC:19;
A8: Cl Int B = Cl B by A3,Def2;
  Int A c= Int Cl B by A5,TOPS_1:19;
  then Cl Int A c= Cl B by A6,PRE_TOPC:19;
  then
A9: Cl B = Cl Int A by A7,A8,XBOOLE_0:def 10;
  Cl B c= Cl A by A4,PRE_TOPC:19;
  then
A10: Int Cl B c= Int Cl A by TOPS_1:19;
  Cl A c= Cl Cl B by A5,PRE_TOPC:19;
  then Int Cl A c= Int Cl Cl B by TOPS_1:19;
  then B = Int Cl A by A6,A10,XBOOLE_0:def 10;
  hence thesis by A4,A5,A9,Th10;
end;
