reserve D for non empty set,
  D1, D2, x, y, Z for set,
  n, k for Nat,
  p, x1, r for Real,
  f for Function,
  Y for RealNormSpace,
  G, H, H1, H2, J for Functional_Sequence of D,the carrier of Y;

theorem
  G + H = H + G & (G + H) + J = G + (H + J)
  proof
    now
      let n be Element of NAT;
      thus (G + H).n = H.n + G.n by Def5
      .= (H + G).n by Def5;
    end;
    hence G + H = H + G by FUNCT_2:63;

    now
      let n be Element of NAT;
      thus ((G + H) + J).n = (G + H).n + J.n by Def5
      .= G.n + H.n + J.n by Def5
      .= G.n + (H.n + J.n) by VFUNCT_1:5
      .= G.n + (H + J).n by Def5
      .= (G + (H + J)).n by Def5;
    end;
    hence thesis by FUNCT_2:63;
  end;
