
theorem EXTh13:
  for G be RealNormSpace-Sequence, p, r be Point of product G, a be Real
  holds
    a * p = r
  iff
    for i be Element of dom G holds r.i = a * (p.i)
  proof
    let G be RealNormSpace-Sequence,
      p,r be Point of product G,
        a be Real;
    reconsider p0 = p, r0 = r as Element of product carr G by EXTh10;
    hereby
      assume
      A1: a * p = r;
      hereby
        let i be Element of dom G;
        reconsider i0 = i as Element of dom carr G by LemmaX;
        A2: (multop G).i0 = the Mult of (G.i0) by PRVECT_2:def 8;
        reconsider rr = a as Element of REAL by XREAL_0:def 1;
        product G = NORMSTR(# product carr G,zeros G,[:addop G:],
                      [:multop G:], productnorm G #) by PRVECT_2:6;
        hence r.i = rr * (p0.i) by A1,A2,PRVECT_2:def 2
        .= a * (p.i);
      end;
    end;
    assume
    A3: for i be Element of dom G holds r.i = a * (p.i);
    reconsider rp = a * p as Element of product carr G by EXTh10;
    A4: ex g be Function
        st rp = g & dom g = dom carr G
         & for i be object st i in dom carr G holds g.i in (carr G).i
        by CARD_3:def 5;
    A5: ex g be Function
        st r0 = g & dom g = dom carr G
         & for i be object st i in dom carr G holds g.i in (carr G).i
        by CARD_3:def 5;
    now
      let i0 be object;
      assume
      A6: i0 in dom rp; then
      reconsider i1 = i0 as Element of dom G by A4,LemmaX;
      reconsider i = i0 as Element of dom carr G by A4,A6;
      A7: product G = NORMSTR(# product carr G,zeros G,[:addop G:],
            [:multop G:], productnorm G #) by PRVECT_2:6;
      reconsider a as Element of REAL by XREAL_0:def 1;
      (multop G).i = the Mult of (G.i) by PRVECT_2:def 8; then
      rp.i0 = a * (p0.i1) by A7,PRVECT_2:def 2;
      hence rp.i0 = r0.i0 by A3;
    end;
    hence a * p = r by A4,A5;
  end;
