
theorem EXTh12:
  for G be RealNormSpace-Sequence,
      p,q,r be Point of product G
  holds
    p+q = r
  iff
    for i be Element of dom G holds r.i = p.i + q.i
  proof
    let G be RealNormSpace-Sequence, p,q,r be Point of product G;
    reconsider p0 = p, q0 = q, r0 = r as Element of product carr G by EXTh10;
    A2: product G = NORMSTR(# product carr G,zeros G,[:addop G:],[:multop G:],
        productnorm G #) by PRVECT_2:6;
    hereby
      assume
      A3: p+q = r;
      hereby
        let i be Element of dom G;
        reconsider i0=i as Element of dom carr G by LemmaX;
        (addop G).i0 = the addF of (G.i0) by PRVECT_1:def 12;
        hence r.i = p.i + q.i by A2,A3,PRVECT_1:def 8;
      end;
    end;
    assume
    A4: for i be Element of dom G holds r.i = p.i + q.i;
    reconsider pq = p+q as Element of product carr G by EXTh10;
    A5: ex g be Function
        st pq = 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;
    A6: 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
      A7: i0 in dom pq; then
      reconsider i1= i0 as Element of dom G by LemmaX,A5;
      reconsider i = i0 as Element of dom carr G by A5,A7;
      (addop G).i = the addF of (G.i) by PRVECT_1:def 12; then
      pq.i0 = p0.i1 + q0.i1 by A2,PRVECT_1:def 8;
      hence pq.i0 = r0.i0 by A4;
    end;
    hence p+q = r by A5,A6;
  end;
