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 Th45:
  P is dense iff for Q st Q <> {} & Q is open holds P meets Q
proof
  hereby
    assume P is dense;
    then
A1: Cl P = [#] TS;
    let Q;
    assume that
A2: Q<>{} and
A3: Q is open;
    set x = the Element of Q;
    x in Q by A2;
    then
A4: TS is non empty;
    x in Cl P by A2,A1,TARSKI:def 3;
    hence P meets Q by A2,A3,A4,Th12;
  end;
  assume
A5: for Q st Q <> {} & Q is open holds P meets Q;
  [#] TS c= Cl P
  proof
    let x be object;
A6: for C be Subset of TS st C is open & x in C holds P meets C by A5;
    assume
A7: x in [#] TS;
    then TS is non empty;
    hence thesis by A7,A6,Th12;
  end;
  then Cl P = [#] TS;
  hence thesis;
end;
