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

theorem Th9:
  A is condensed implies Int Cl A = Int A & Cl A = Cl Int A
proof
A1: Cl Int A c= Cl A by PRE_TOPC:19,TOPS_1:16;
  assume
A2: A is condensed;
  then Int Cl A c= A by TOPS_1:def 6;
  then
A3: Int Int Cl A c= Int A by TOPS_1:19;
  A c= Cl Int A by A2,TOPS_1:def 6;
  then
A4: Cl A c= Cl Cl Int A by PRE_TOPC:19;
  Int A c= Int Cl A by PRE_TOPC:18,TOPS_1:19;
  hence thesis by A3,A4,A1;
end;
