
theorem TOPS145:
  for X be RealNormSpace, R be Subset of X
  holds
    R is dense
  iff
    for S be Subset of X st S <> {} & S is open holds R meets S
  proof
    let X be RealNormSpace, R be Subset of X;
    reconsider R1 = R as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
    hereby
      assume R is dense; then
      A1: R1 is dense by EQCL2;
      thus for S be Subset of X st S <> {} & S is open holds R meets S
      proof
        let S be Subset of X;
        assume
        A2: S <> {} & S is open;
        reconsider S1 = S as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
        S1 is open by A2,NORMSP_2:33;
        hence R meets S by A1,A2,TOPS_1:45;
      end;
    end;
    assume
    A3: for S be Subset of X st S <> {} & S is open holds R meets S;
    for S1 be Subset of LinearTopSpaceNorm X st S1 <> {} & S1 is open
    holds R1 meets S1
    proof
      let S1 be Subset of LinearTopSpaceNorm X;
      assume
      A4: S1 <> {} & S1 is open;
      reconsider S = S1 as Subset of X by NORMSP_2:def 4;
      S is open by A4,NORMSP_2:33;
      hence R1 meets S1 by A3,A4;
    end; then
    R1 is dense by TOPS_1:45;
    hence R is dense by EQCL2;
  end;
