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

theorem :: Theorem 4
  A is condensed iff ex B st B is regular_closed & Int B c= A & A c= B
proof
  thus A is condensed implies ex B st B is regular_closed & Int B c= A & A c= B
  proof
    assume
A1: A is condensed;
    then
A2: Cl Int A = Cl A;
    take Cl Int 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_closed and
A4: Int B c= A and
A5: A c= B;
A6: Cl Int B = B by A3,TOPS_1:def 7;
  Cl A c= Cl B by A5,PRE_TOPC:19;
  then Int Cl A c= Int Cl B by TOPS_1:19;
  then
A7: Int Cl A c= Int B by A3,Def1;
  Cl Int B c= Cl A by A4,PRE_TOPC:19;
  then Int B c= Int Cl A by A6,TOPS_1:19;
  then
A8: Int B = Int Cl A by A7,XBOOLE_0:def 10;
  Int A c= Int B by A5,TOPS_1:19;
  then
A9: Cl Int A c= Cl Int B by PRE_TOPC:19;
  Int Int B c= Int A by A4,TOPS_1:19;
  then Cl Int Int B c= Cl Int A by PRE_TOPC:19;
  then Cl Int A = B by A6,A9,XBOOLE_0:def 10;
  hence thesis by A4,A5,A8,Th10;
end;
