
theorem
  for X be RealNormSpace,
      Y be Subset of X,
      v be object
  st v in the carrier of X
  holds
    v in Cl(Y)
  iff
    for G being Subset of X
    st G is open & v in G
    holds G meets Y
  proof
    let X be RealNormSpace,
        Y be Subset of X,
        v be object;
    assume
    A1: v in the carrier of X;
    reconsider Z = Y as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
    hereby
      assume v in Cl(Y); then
      A2: v in Cl(Z) by EQCL1;
      thus for G being Subset of X st G is open & v in G holds G meets Y
      proof
        let G be Subset of X;
        assume
        A3: G is open & v in G;
        reconsider G0 = G as Subset of LinearTopSpaceNorm X
        by NORMSP_2:def 4;
        G0 is open by A3,NORMSP_2:33;
        hence G meets Y by A2,A3,PRE_TOPC:def 7;
      end;
    end;
    assume
    A4: for G being Subset of X st G is open & v in G holds G meets Y;
    A5: for G0 being Subset of LinearTopSpaceNorm X
    st G0 is open & v in G0 holds G0 meets Z
    proof
      let G0 be Subset of LinearTopSpaceNorm X;
      assume
      A6: G0 is open & v in G0;
      reconsider G = G0 as Subset of X by NORMSP_2:def 4;
      G is open by A6,NORMSP_2:33;
      hence G0 meets Z by A4,A6;
    end;
    v in the carrier of LinearTopSpaceNorm X by A1,NORMSP_2:def 4; then
    v in Cl(Z) by A5,PRE_TOPC:def 7;
    hence v in Cl(Y) by EQCL1;
  end;
