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 open & R is condensed implies R is open_condensed) & (P is
  open_condensed implies P is open & P is condensed)
proof
  hereby
    assume that
A1: R is open and
A2: R is condensed;
    R = Int R by A1,Th23;
    then
A3: R c= Int(Cl R) by Th19,PRE_TOPC:18;
    Int Cl R c= R by A2;
    then Int Cl R = R by A3;
    hence R is open_condensed;
  end;
  assume
A4: P is open_condensed;
  then
A5: Fr Cl P = Cl P \ P by Th64;
  Fr P = Fr Cl P by A4;
  then P is open by A5,Th42;
  then Int P = P by Th23;
  then
A6: P c= Cl Int P by PRE_TOPC:18;
  P = Int(Cl P) by A4;
  hence thesis by A6;
end;
