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 closed & R is condensed implies R is closed_condensed) &
  (P is closed_condensed implies P is closed & P is condensed)
proof
  hereby
    assume that
A1: R is closed and
A2: R is condensed;
A3: R = Cl R by A1,PRE_TOPC:22;
    then Int R c= R by A2;
    then
A4: Cl(Int R) c= R by A3,PRE_TOPC:19;
    R c= Cl(Int R) by A2;
    then Cl(Int R) = R by A4;
    hence R is closed_condensed;
  end;
  assume
A5: P is closed_condensed;
  Fr(Int P) = Cl(Int P) \ Int(Int P) by Lm2;
  then Fr P = Cl(Int P) \ Int(Int P) by A5
    .= P \ Int P by A5;
  then P is closed by Th43;
  then Cl P = P by PRE_TOPC:22;
  then
A6: Int Cl P c= P by Th16;
  Cl Int P = P by A5;
  hence thesis by A6;
end;
