reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;

theorem
  A is dense iff for G being Subset of X holds G is open implies Cl G =
  Cl(G /\ A)
proof
  thus A is dense implies for G being Subset of X holds G is open implies Cl G
  = Cl(G /\ A)
  proof
    assume
A1: A is dense;
    let G be Subset of X;
    assume G is open;
    then
A2: Cl(G /\ Cl A) = Cl(G /\ A) by TOPS_1:14;
    G /\ [#]X = G by XBOOLE_1:28;
    hence thesis by A1,A2;
  end;
  assume for G being Subset of X holds G is open implies Cl G = Cl(G /\ A);
  then Cl [#]X = Cl([#]X /\ A);
  then
A3: [#]X = Cl([#]X /\ A) by TOPS_1:2;
  [#]X /\ A = A by XBOOLE_1:28;
  hence thesis by A3;
end;
