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
  Q is open implies Cl(Q /\ Cl K) = Cl(Q /\ K)
proof
A1: Cl (Q /\ K) c= Cl(Q /\ Cl K)
  proof
    let x be object;
    assume
A2: x in Cl(Q /\ K);
    then reconsider p99= x as Point of TS;
A3: TS is non empty by A2;
    for Q1 being Subset of TS holds Q1 is open implies (p99 in Q1 implies
    (Q /\ Cl K) meets Q1)
    proof
      let Q1 be Subset of TS;
      assume
A4:   Q1 is open;
      assume p99 in Q1;
      then (Q /\ K) meets Q1 by A2,A3,A4,Th12;
      then
A5:   (Q /\ K) /\ Q1 <> {};
      Q /\ K c= Q /\ Cl K by PRE_TOPC:18,XBOOLE_1:26;
      then (Q /\ Cl K) /\ Q1 <> {} by A5,XBOOLE_1:3,26;
      hence thesis;
    end;
    hence thesis by A3,Th12;
  end;
  assume Q is open;
  then Cl(Q /\ Cl K) c= Cl(Cl(Q /\ K)) by Th13,PRE_TOPC:19;
  hence thesis by A1;
end;
