reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;

theorem Th7:
  A is condensed & Cl Int A c= Int Cl A implies A is open_condensed
  & A is closed_condensed
proof
  assume
A1: A is condensed;
  then
A2: Int Cl A c= A by TOPS_1:def 6;
A3: A c= Cl Int A by A1,TOPS_1:def 6;
  assume
A4: Cl Int A c= Int Cl A;
  then Cl Int A c= A by A2;
  then
A5: A = Cl Int A by A3;
  A c= Int Cl A by A3,A4;
  then A = Int Cl A by A2;
  hence thesis by A5,TOPS_1:def 7,def 8;
end;
