reserve G for RealNormSpace-Sequence;

theorem Th9:
  for x be Element of product carr G, i be Nat st
    i in dom x holds 0 <= normsequence(G,x).i
proof
  let x be Element of product carr G, i be Nat such that
A1: i in dom x;
  dom G = Seg len G by FINSEQ_1:def 3
    .= Seg len carr G by PRVECT_1:def 11
    .= dom carr G by FINSEQ_1:def 3;
  then reconsider i0=i as Element of dom G by A1,CARD_3:9;
  dom x = dom carr G & (carr G).i0 = the carrier of G.i0
    by PRVECT_1:def 11,CARD_3:9;
  then reconsider xi0=x.i0 as Element of G.i0 by A1,CARD_3:9;
  0<= ||.xi0.|| by NORMSP_1:4;
  hence thesis by Def11;
end;
