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 Th13:
  Q is open implies Q /\ Cl K c= Cl(Q /\ K)
proof
  assume
A1: Q is open;
  let x be object;
  assume
A2: x in Q /\ Cl K;
  then
A3: x in Cl K by XBOOLE_0:def 4;
  reconsider p99= x as Point of TS by A2;
A4: TS is non empty by A2;
A5: x in Q by A2,XBOOLE_0:def 4;
  for Q1 being Subset of TS holds Q1 is open implies (p99 in Q1 implies (Q
  /\ K) meets Q1)
  proof
    let Q1 be Subset of TS;
    assume
A6: Q1 is open;
    assume p99 in Q1;
    then p99 in Q1 /\ Q by A5,XBOOLE_0:def 4;
    then K meets (Q1 /\ Q) by A1,A3,A4,A6,Th12;
    then
A7: K /\ (Q1 /\ Q) <> {};
    K /\ (Q1 /\ Q) = (Q /\ K) /\ Q1 by XBOOLE_1:16;
    hence thesis by A7;
  end;
  hence thesis by A4,Th12;
end;
