reserve TS for 1-sorted,
  K, Q for Subset of TS;
reserve TS for TopSpace,
  GX for TopStruct,
  x for set,
  P, Q for Subset of TS,
  K , L for Subset of TS,
  R, S for Subset of GX,
  T, W for Subset of GX;

theorem
  R is condensed implies Int R is condensed & Cl R is condensed
proof
  Cl(Int R) c= Cl R by Th16,PRE_TOPC:19;
  then
A1: Int(Cl(Int R)) c= Int(Cl R) by Th19;
A2: R c= Cl R by PRE_TOPC:18;
  then (Cl R)` c= R` by SUBSET_1:12;
  then Cl((Cl R)`) c= Cl(R`) by PRE_TOPC:19;
  then (Cl R`)` c= (Cl(((Cl R)`)))` by SUBSET_1:12;
  then
A3: Cl(Int R) c= Cl((Cl((Cl R)`))`) by PRE_TOPC:19;
  assume
A4: R is condensed;
  then
A5: R c= Cl(Int R);
  then Cl R c= Cl(Cl(Int R)) by PRE_TOPC:19;
  then
A6: Cl R c= Cl(Int(Cl R)) by A3;
A7: Int(Cl R) c= R by A4;
  then Int(Int(Cl R)) c= Int R by Th19;
  then
A8: Int(Cl(Int R)) c= Int R by A1;
  Int R c= R by Th16;
  then
A9: Int R c= Cl(Int(Int R)) by A5;
  (Int(Cl(Cl R))) c= Cl R by A7,A2;
  hence thesis by A9,A6,A8;
end;
