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 Th46:
  P is dense implies for Q holds Q is open implies Cl Q = Cl(Q /\ P)
proof
  assume
A1: P is dense;
  let Q;
  assume
A2: Q is open;
  thus Cl Q c= Cl(Q /\ P)
  proof
    let x be object;
    assume
A3: x in Cl Q;
    then
A4: TS is non empty;
    now
      let Q1 be Subset of TS;
      assume
A5:   Q1 is open;
      assume x in Q1;
      then Q meets Q1 by A3,A4,A5,Th12;
      then Q /\ Q1 <> {};
      then P meets (Q /\ Q1) by A1,A2,A5,Th45;
      then
A6:   P /\ (Q /\ Q1) <> {};
      P /\ (Q /\ Q1) = (Q /\ P) /\ Q1 by XBOOLE_1:16;
      hence (Q /\ P) meets Q1 by A6;
    end;
    hence thesis by A3,A4,Th12;
  end;
  thus thesis by PRE_TOPC:19,XBOOLE_1:17;
end;
