
theorem
  for G be RealNormSpace-Sequence,
      p be Point of product G
  holds
    0.(product G) = p
  iff
    for i be Element of dom G holds p.i = 0.(G.i)
  proof
    let G be RealNormSpace-Sequence,
        p be Point of product G;
    reconsider p0 = p as Element of product carr G by EXTh10;
    A1: dom carr G = dom G by LemmaX;
    A2: product G = NORMSTR(# product carr G,zeros G,[:addop G:],
        [:multop G:], productnorm G #) by PRVECT_2:6;
    hence 0.(product G) = p
      implies for i be Element of dom G holds p.i = 0.(G.i)
        by A1,PRVECT_1:def 14;
    assume
    A3: for i be Element of dom G holds p.i = 0.(G.i);
    now
      let i0 be Element of dom carr G;
      reconsider i = i0 as Element of dom G by LemmaX;
      p0.i0 = 0.(G.i) by A3;
      hence p0.i0 = 0.(G.i0);
    end;
    hence 0.(product G)=p by A2,PRVECT_1:def 14;
  end;
