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 Th3:
  H1 = G - H iff for n holds H1.n = G.n - H.n
  proof
    thus H1 = G - H implies for n holds H1.n = G.n - H.n
    proof
      assume
      A1: H1 = G - H;
      let n;
      thus H1.n = G.n + (-H).n by A1, Def5
      .= G.n + - H.n by Def3
      .= G.n -H.n by VFUNCT_1:25;
    end;
    assume
    A2: for n holds H1.n = G.n - H.n;
    now
      let n be Element of NAT;
      thus H1.n = G.n - H.n by A2
      .= G.n + (-H.n) by VFUNCT_1:25
      .= G.n + (-H).n by Def3
      .= (G - H).n by Def5;
    end;
    hence thesis by FUNCT_2:63;
  end;
