reserve G for RealNormSpace-Sequence;

theorem
  for G be RealNormSpace-Sequence, seq be sequence of product G,
      x0 be Point of product G, y0 be Element of product carr G st
    x0 = y0 & seq is convergent & lim seq=x0 holds
    for i be Element of dom G ex seqi be sequence of
G.i st seqi is convergent & y0.i = lim seqi & for m be Element of NAT holds ex
  seqm be Element of product carr G st seqm= seq.m & seqi.m=seqm.i
proof
  let G be RealNormSpace-Sequence, seq be sequence of (product G),
      x0 be Point of product G, y0 be Element of product carr G;
  assume that
A1: x0 = y0 and
A2: seq is convergent & lim seq=x0;
  let i be Element of dom G;
  defpred P1[Nat,Element of G.i] means ex seqm be Element of
  product carr G st seqm = seq.$1 & $2 = seqm.i;
  len G = len carr G by PRVECT_1:def 11;
  then
A3: dom G = dom carr G by FINSEQ_3:29;
  then y0.i in (carr G).i by CARD_3:9;
  then reconsider x0i=y0.i as Point of G.i by PRVECT_1:def 11;
A4: for x being Element of NAT ex y being Element of G.i st P1[x,y]
  proof
    let x be Element of NAT;
    product G = NORMSTR(# product carr G,zeros G,[:addop G:], [:multop G:]
      ,productnorm G #) by Th6;
    then consider seqm be Element of product carr G such that
A5: seqm =seq.x;
    take seqm.i;
    (carr G).i = the carrier of G.i by PRVECT_1:def 11;
    hence thesis by A3,A5,CARD_3:9;
  end;
  consider f be sequence of the carrier of G.i such that
A6:  for x being Element of NAT holds P1[x,f.x] from FUNCT_2:sch 3(A4);
 for x being Nat holds P1[x,f.x]
  proof let x be Nat;
    x in NAT by ORDINAL1:def 12;
   hence thesis by A6;
  end;
  then consider seqi be sequence of G.i such that
A7: for m be Nat ex seqm be Element of product carr G
  st seqm= seq.m & seqi.m=seqm.i;
  take seqi;
A8: for r be Real
    st 0 < r ex m be Nat st
   for n be Nat st m <= n holds ||.seqi.n - x0i.|| < r
  proof
    let r be Real;
    assume r > 0;
    then consider k be Nat such that
A9: for n be Nat st k <= n holds ||.seq.n - x0.|| < r by A2,NORMSP_1:def 7;
    take k;
      let n be Nat;
      assume n >= k; then
A10:   ||.(seq.n) - x0.|| < r by A9;
      ex seqn be Element of product carr G st seqn = seq.n & seqi.n =
      seqn.i by A7;
      then ||. seqi.n - x0i.|| <= ||. seq.n - x0.|| by A1,Th11;
      hence ||.(seqi.n) - x0i.|| < r by A10,XXREAL_0:2;
  end;
  then seqi is convergent;
  hence thesis by A7,A8,NORMSP_1:def 7;
end;
