reserve G for RealNormSpace-Sequence;

theorem Th13:
  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 &
  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 holds seq is convergent & lim seq=x0
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: 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;
  defpred PP[Element of dom G, set] means ex rseqi be Real_Sequence, seqi be
  sequence of G.$1 st rseqi=$2 & seqi is convergent & rseqi is convergent & lim
rseqi=0 & rseqi=||.seqi-lim seqi.|| & for m be Element of NAT holds ex seqm be
  Element of product carr G st seqm= seq.m & seqi.m=seqm.$1;
A3: for i be Element of dom G ex yyseqi be Element of Funcs(NAT,REAL) st PP[
  i,yyseqi]
  proof
    let i be Element of dom G;
    consider seqi be sequence of G.i such that
A4: seqi is convergent and
    y0.i = lim seqi and
A5: for m be Element of NAT holds ex Sm be Element of product carr G
    st Sm= seq.m & seqi.m=Sm.i by A2;
    set rseqi=||.seqi-lim seqi.||;
A6: rseqi is Element of Funcs(NAT,REAL) by FUNCT_2:8;
    rseqi is convergent & lim rseqi=0 by A4,Lm7;
    hence thesis by A4,A5,A6;
  end;
  consider yyseq be Function of (dom G),Funcs(NAT,REAL) such that
A7: for i be Element of dom G holds PP[i,yyseq.i] from FUNCT_2:sch 3(A3);
  reconsider I = len G as Element of NAT;
  defpred PQ[Element of NAT,Element of (REAL I)] means for i be Element of dom
  G holds (yyseq.i).$1 = $2.i;
A8: for k be Element of NAT ex yseqk be Element of (REAL I) st PQ[k,yseqk]
  proof
    let k be Element of NAT;
    defpred PPF[set,object] means
ex i be Element of dom G st i=$1 & $2 = (yyseq.i).k;
A9: for k be Nat st k in Seg I ex x be object st PPF[k,x]
    proof
      let j be Nat;
      assume j in Seg I;
      then reconsider i=j as Element of dom G by FINSEQ_1:def 3;
      take (yyseq.i).k;
      thus thesis;
    end;
    consider yseqk be FinSequence such that
A10: dom yseqk = Seg I & for j be Nat st j in Seg I holds PPF[j,yseqk.
    j] from FINSEQ_1:sch 1(A9);
    now
      let j be Nat;
      assume j in dom yseqk;
      then consider i be Element of dom G such that
      i=j and
A11:  yseqk.j= (yyseq.i).k by A10;
      yyseq.i is sequence of REAL by FUNCT_2:66;
      hence yseqk.j in REAL by A11,FUNCT_2:5;
    end;
    then reconsider yyy=yseqk as FinSequence of REAL by FINSEQ_2:12;
    yyy is Element of (len yyy)-tuples_on REAL by FINSEQ_2:92;
    then reconsider yseqk as Element of (REAL I) by A10,FINSEQ_1:def 3;
    now
      let j be Element of dom G;
      j in dom G;
      then j in Seg I by FINSEQ_1:def 3;
      then ex i be Element of dom G st i=j & yseqk.j= (yyseq.i).k by A10;
      hence yseqk.j= (yyseq.j).k;
    end;
    hence thesis;
  end;
  consider yseq be sequence of REAL I such that
A12: for k be Element of NAT holds PQ[k,yseq.k] from FUNCT_2:sch 3(A8);
A13: now
    let i0 be Element of NAT;
    assume
 i0 in Seg I;
    then reconsider i=i0 as Element of dom G by FINSEQ_1:def 3;
    take i;
    consider rseqi be Real_Sequence, seqi be sequence of G.i such that
A14: rseqi=yyseq.i & seqi is convergent & rseqi is convergent & lim
rseqi= 0 &( rseqi=||.seqi-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) by A7;
    take rseqi,seqi;
    thus (for k be Element of NAT holds rseqi.k = (yseq.k).i0) & i=i0 & seqi
    is convergent & rseqi is convergent & lim rseqi=(0*I).i & rseqi=||.seqi-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 by A12,A14;
  end;
  reconsider xseq=yseq as sequence of REAL-NS I by REAL_NS1:def 4;
  xseq = yseq;
  then consider
  xseq be sequence of REAL-NS I, yseq be sequence of REAL I such that
A15: xseq=yseq and
A16: for i0 be Element of NAT st i0 in Seg I ex i be Element of dom G,
  rseqi be Real_Sequence, seqi be sequence of G.i st (for k be Element of NAT
holds rseqi.k = (yseq.k).i0) & i=i0 & seqi is convergent & rseqi is convergent
& lim rseqi=(0*I).i & rseqi=||.seqi-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 by A13;
A17: for i be Nat st i in Seg I ex rseqi be Real_Sequence st (for
k be Nat holds rseqi.k = (yseq.k).i) & rseqi is convergent & (0*I).i
  = lim rseqi
  proof
    let i0 be Nat;
    assume i0 in Seg I;
    then consider i be Element of dom G,
     rseqi be Real_Sequence, seqi be sequence of G.i such that
A18: for k be Element of NAT holds rseqi.k = (yseq.k).i0 and
A19:   i=i0 & seqi is convergent & rseqi is convergent &
     lim rseqi=(0*I).i & rseqi=||.seqi-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 by A16;
    take rseqi;
    thus for k be Nat holds rseqi.k = (yseq.k).i0
     proof let k be Nat;
       k in NAT by ORDINAL1:def 12;
      hence thesis by A18;
     end;
   thus thesis by A19;
  end;
A20: product G = NORMSTR(# product carr G,zeros G, [:addop G:],[:multop G:],
    productnorm G #) by Th6;
  now
    let x be object;
    assume x in NAT;
    then reconsider i=x as Element of NAT;
    reconsider seqimx0 = seq.i-x0 as Element of product carr G by A20;
A21: now
      reconsider mx0=-x0 as Element of product carr G by A20;
      let k be Nat;
      assume
A22:  k in dom normsequence(G,seqimx0);
A23:  len G= len normsequence(G,seqimx0) by Def11;
      then reconsider k0=k as Element of dom G by A22,FINSEQ_3:29;
      k in Seg I by A22,A23,FINSEQ_1:def 3;
      then consider
      k1 be Element of dom G, rseqk1i be Real_Sequence, seqk1i be
      sequence of G.k1 such that
A24:  for j be Element of NAT holds rseqk1i.j = (yseq.j).k and
A25:  k1=k and
      seqk1i is convergent and
      rseqk1i is convergent and
      lim rseqk1i=(0*I).k1 and
A26:  rseqk1i=||.seqk1i-lim seqk1i.|| and
A27:  for m be Element of NAT holds ex seqk1m be Element of product
      carr G st seqk1m= seq.m & seqk1i.m=seqk1m.k1 by A16;
      consider seqk0 be sequence of G.k0 such that
      seqk0 is convergent and
A28:  y0.k0 = lim seqk0 and
A29:  for m be Element of NAT holds ex seqm0 be Element of product
      carr G st seqm0= seq.m & seqk0.m=seqm0.k0 by A2;
A30:  ex seqm0 be Element of product carr G st seqm0= seq.i & seqk0.i=
      seqm0.k0 by A29;
      now
        let x be object;
        assume x in NAT;
        then reconsider m = x as Element of NAT;
        ( ex seqk1m be Element of product carr G st seqk1m= seq.m &
seqk1i.m=seqk1m.k1) & ex seqm0 be Element of product carr G st seqm0= seq.m &
        seqk0.m = seqm0.k0 by A29,A27;
        hence seqk1i.x = seqk0.x by A25;
      end;
      then
A31:  seqk1i=seqk0 by A25,FUNCT_2:12;
      len G = len carr G by PRVECT_1:def 11;
      then
A32:  dom G = dom carr G by FINSEQ_3:29;
      -x0 = (-1) * x0 by RLVECT_1:16;
      then mx0.k0 = (-jj)*(lim seqk0) by A1,A20,A28,A32,Lm4;
      then -(lim seqk0) = mx0.k0 by RLVECT_1:16;
      then
A33:  seqimx0.k0 = seqk0.i - lim seqk0 by A20,A30,A32,Lm3;
      thus (normsequence(G,seqimx0)).k = (the normF of G.k0).(seqimx0.k0) by
Def11
        .= ||.(seqk0 -lim seqk0).i .|| by A33,NORMSP_1:def 4
        .= (||.seqk1i - (lim seqk1i).||).i by A25,A31,NORMSP_0:def 4
        .= (yseq.i).k by A24,A26;
    end;
    len (yseq.i)= I by CARD_1:def 7;
    then len (yseq.i) = len normsequence(G,seqimx0) by Def11;
    then dom (yseq.i) = dom normsequence(G,seqimx0) by FINSEQ_3:29;
    then
A34: yseq.i = normsequence(G,seqimx0) by A21,FINSEQ_1:13;
    thus ||.xseq-0.(REAL-NS I).||.x =||.(xseq-0.(REAL-NS I)).i .|| by
NORMSP_0:def 4
      .=||.xseq.i-0.(REAL-NS I).|| by NORMSP_1:def 4
      .=||.xseq.i.||
      .=|.yseq.i.| by A15,REAL_NS1:1
      .=||.seq.i -x0.|| by A34,Th7
      .=||.(seq -x0).i.|| by NORMSP_1:def 4
      .=||.seq-x0.||.x by NORMSP_0:def 4;
  end;
  then
A35: ||.xseq-0.(REAL-NS I).|| =||.seq-x0.|| by FUNCT_2:12;
  (0*I) = 0.(REAL-NS I) by REAL_NS1:def 4;
  then xseq is convergent & lim xseq = 0.(REAL-NS I) by A15,A17,REAL_NS1:11;
  then ||.seq-x0.|| is convergent & lim ||.seq-x0.|| =0 by A35,Lm7;
  hence thesis by Lm7;
end;
