reserve NRM for non empty RealNormSpace;
reserve seq for sequence of NRM;

theorem
  for vseq be sequence of l1_Space st vseq is Cauchy_sequence_by_Norm
  holds vseq is convergent
proof
  let vseq be sequence of l1_Space such that
A1: vseq is Cauchy_sequence_by_Norm;
  defpred P[object,object] means ex i be Nat st $1=i & ex rseqi be
  Real_Sequence st (for n be Nat holds rseqi.n=(seq_id(vseq.n)).i) &
  rseqi is convergent & $2 = lim rseqi;
A2: for x be object st x in NAT ex y be object st y in REAL & P[x,y]
  proof
    let x be object;
    assume x in NAT;
    then reconsider i=x as Nat;
    deffunc F(Nat) = (seq_id(vseq.$1)).i;
    consider rseqi be Real_Sequence such that
A3: for n be Nat holds rseqi.n= F(n) from SEQ_1:sch 1;
     reconsider lr = lim rseqi as Element of REAL by XREAL_0:def 1;
    take lr;
    now
      let e be Real such that
A4:   e > 0;
      thus ex k be Nat st for m be Nat st k <= m holds
      |.rseqi.m -rseqi.k.| < e
      proof
        consider k be Nat such that
A5:     for n, m be Nat st n >= k & m >= k holds ||.(vseq.n
        ) - (vseq.m).|| < e by A1,A4,Th8;
        for m being Nat st k <= m holds |.rseqi.m-rseqi.k.| < e
        proof
          let m be Nat such that
A6:       k<=m;
A7:       for i be Nat holds 0 <= |.seq_id((vseq.m) - (vseq.
          k)).|. i
          proof
            let i be Nat;
            0 <= |.(seq_id((vseq.m) - (vseq.k))).i.| by COMPLEX1:46;
            hence thesis by SEQ_1:12;
          end;
          seq_id((vseq.m)-(vseq.k)) is absolutely_summable by Def1;
          then abs(seq_id((vseq.m)-(vseq.k))) is summable by SERIES_1:def 4;
          then
A8:       abs(seq_id((vseq.m) - (vseq.k))).i <= Sum(|.seq_id((vseq.m) -
          (vseq.k)).|) by A7,RSSPACE2:3;
          seq_id((vseq.m) - (vseq.k)) =seq_id(seq_id((vseq.m))-seq_id((
          vseq.k))) by Th6
            .= seq_id((vseq.m))+-seq_id((vseq.k)) by SEQ_1:11;
          then
          (seq_id((vseq.m) - (vseq.k))).i =(seq_id((vseq.m))).i+(-seq_id(
          (vseq.k))).i by SEQ_1:7
            .=(seq_id((vseq.m))).i+(-(seq_id((vseq.k))).i) by SEQ_1:10
            .=(seq_id((vseq.m))).i-(seq_id((vseq.k))).i
            .=rseqi.m -(seq_id((vseq.k))).i by A3
            .=rseqi.m - rseqi.k by A3;
          then
A9:       |.rseqi.m-rseqi.k.| = abs(seq_id((vseq.m) - (vseq.k))).i by SEQ_1:12;
          ||.(vseq.m) - (vseq.k).|| =Sum(|.seq_id((vseq.m) - (vseq.k)).|
          ) by Th6;
          then Sum(|.seq_id((vseq.m) - (vseq.k)).|) < e by A5,A6;
          hence thesis by A8,A9,XXREAL_0:2;
        end;
        hence thesis;
      end;
    end;
    then rseqi is convergent by SEQ_4:41;
    hence thesis by A3;
  end;
  consider f be sequence of REAL such that
A10: for x be object st x in NAT holds P[x,f.x] from FUNCT_2:sch 1(A2);
  reconsider tseq=f as Real_Sequence;
A11: now
    let i be Nat;
    reconsider x=i as set;
    i in NAT by ORDINAL1:def 12;
    then ex i0 be Nat st x=i0 & ex rseqi be Real_Sequence st ( for
n be Nat holds rseqi.n=(seq_id(vseq.n)).i0 ) & rseqi is convergent &
    f.x=lim rseqi by A10;
    hence
    ex rseqi be Real_Sequence st ( for n be Nat holds rseqi.n=
    (seq_id(vseq.n)).i ) & rseqi is convergent & tseq.i=lim rseqi;
  end;
A12: for e be Real
   st e >0 ex k be Nat st for n be Nat
  st n >= k holds abs(seq_id tseq-seq_id(vseq.n)) is summable & Sum(abs(seq_id
  tseq-seq_id(vseq.n))) < e
  proof
    let e1 be Real such that
A13: e1 >0;
    reconsider e1 as Real;
    set e=e1/2;
A14: e < e1 by A13,XREAL_1:216;
    e > 0 by A13,XREAL_1:215;
    then consider k be Nat such that
A15: for n, m be Nat st n >= k & m >= k holds ||.(vseq.n) -
    (vseq.m).|| < e by A1,Th8;
A16: for m,n be Nat st n >= k & m >= k holds ( abs(seq_id((vseq
.n) - (vseq.m))) is summable & Sum(abs(seq_id((vseq.n) - (vseq.m)))) < e & for
    i be Nat holds 0 <= abs(seq_id((vseq.n) - (vseq.m))).i )
    proof
      let m,n be Nat;
      assume n >= k & m >= k;
      then ||.(vseq.n) - (vseq.m).|| < e by A15;
      then
A17:  (the normF of l1_Space).((vseq.n) - (vseq.m)) < e;
A18:  for i be Nat holds 0 <= abs(seq_id((vseq.n) - (vseq.m))) .i
      proof
        let i be Nat;
        0 <= |.(seq_id((vseq.n) - (vseq.m))).i.| by COMPLEX1:46;
        hence thesis by SEQ_1:12;
      end;
      seq_id(((vseq.n) - (vseq.m))) is absolutely_summable by Def1;
      hence thesis by A17,A18,Def2,SERIES_1:def 4;
    end;
A19: for n be Nat for i be Nat holds for rseq be
Real_Sequence st ( for m be Nat holds rseq.m=Partial_Sums(abs(seq_id
((vseq.m)-(vseq.n)))).i ) holds rseq is convergent & lim rseq = Partial_Sums(
    abs(seq_id tseq-seq_id(vseq.n))).i
    proof
      let n be Nat;
      defpred P[Nat] means
for rseq be Real_Sequence st for m be
Nat holds rseq.m=Partial_Sums(abs(seq_id((vseq.m) - (vseq.n)))).$1
holds rseq is convergent & lim rseq = Partial_Sums(abs(seq_id tseq-seq_id(vseq.
      n))).$1;
A20:  for m,k be Nat holds seq_id((vseq.m) - (vseq.k)) =
      seq_id((vseq.m))-seq_id((vseq.k))
      proof
        let m,k be Nat;
        seq_id((vseq.m) - (vseq.k)) = seq_id(seq_id((vseq.m))-seq_id((
        vseq.k))) by Th6;
        hence thesis;
      end;
      now
        let i be Nat such that
A21:    for rseq be Real_Sequence st ( for m be Nat holds
rseq.m= Partial_Sums(abs(seq_id((vseq.m) - (vseq.n)))).i ) holds rseq is
convergent & lim rseq = Partial_Sums(abs((seq_id tseq - seq_id(vseq.n)))).i;
        thus for rseq be Real_Sequence st ( for m be Nat holds rseq
.m = Partial_Sums(abs(seq_id((vseq.m) - (vseq.n)))).(i+1) ) holds rseq is
convergent & lim rseq =Partial_Sums(abs((seq_id tseq - seq_id(vseq.n)))).(i+1)
        proof
          deffunc F(Nat) = Partial_Sums(abs(seq_id((vseq.$1) - (
          vseq.n)))).i;
          consider rseqb be Real_Sequence such that
A22:      for m be Nat holds rseqb.m = F(m) from SEQ_1:sch 1;
          consider rseq0 be Real_Sequence such that
A23:      for m be Nat holds rseq0.m=(seq_id(vseq.m)).(i+1 ) and
A24:      rseq0 is convergent and
A25:      tseq.(i+1)=lim rseq0 by A11;
          let rseq be Real_Sequence such that
A26:      for m be Nat holds rseq.m = Partial_Sums(abs(
          seq_id((vseq.m) - (vseq.n)))).(i+1);
A27:      now
            let m be Nat;
            thus rseq.m = Partial_Sums(abs(seq_id((vseq.m) - (vseq.n)))).(i+1)
            by A26
              .=abs(seq_id((vseq.m) - (vseq.n))).(i+1) +Partial_Sums(abs(
            seq_id((vseq.m) - (vseq.n)))).i by SERIES_1:def 1
              .=abs(seq_id(vseq.m)-seq_id(vseq.n)).(i+1) +Partial_Sums(abs(
            seq_id((vseq.m) - (vseq.n)))).i by A20
              .=abs(seq_id(vseq.m)-seq_id(vseq.n)).(i+1) + rseqb.m by A22
              .=|.(seq_id(vseq.m)-seq_id(vseq.n)).(i+1).| + rseqb.m by
SEQ_1:12
              .=|.(seq_id(vseq.m)+-seq_id(vseq.n)).(i+1).| + rseqb.m by
SEQ_1:11
              .=|.(seq_id(vseq.m)).(i+1)+(-seq_id(vseq.n)).(i+1).| + rseqb.
            m by SEQ_1:7
              .=|.(seq_id(vseq.m)).(i+1)+-(seq_id(vseq.n)).(i+1).| + rseqb.
            m by SEQ_1:10
              .= |.(seq_id(vseq.m)).(i+1)-(seq_id(vseq.n)).(i+1).| + rseqb.
            m
              .= |.rseq0.m-(seq_id(vseq.n)).(i+1).| + rseqb.m by A23;
          end;
A28:      rseqb is convergent by A21,A22;
          then lim rseq = |.lim(rseq0)-(seq_id(vseq.n)).(i+1).| + lim rseqb
          by A24,A27,Lm3
            .= |.tseq.(i+1)+-(seq_id(vseq.n)).(i+1).| + lim rseqb by A25
            .= |.tseq.(i+1)+(-seq_id(vseq.n)).(i+1).| + lim rseqb by SEQ_1:10
            .= |.(tseq+(-seq_id(vseq.n))).(i+1).| + lim rseqb by SEQ_1:7
            .= |.(tseq-(seq_id(vseq.n))).(i+1).| + lim rseqb by SEQ_1:11
            .= abs(tseq-(seq_id(vseq.n))).(i+1) + lim rseqb by SEQ_1:12
            .= abs(tseq-(seq_id(vseq.n))).(i+1) + Partial_Sums(abs((seq_id
          tseq -seq_id(vseq.n)))).i by A21,A22
            .= Partial_Sums (abs((seq_id tseq -seq_id(vseq.n)))).(i+1) by
SERIES_1:def 1;
          hence thesis by A28,A24,A27,Lm3;
        end;
      end;
      then
A29:  for i be Nat st P[i] holds P[i+1];
      now
        let rseq be Real_Sequence such that
A30:    for m be Nat holds rseq.m=Partial_Sums(abs(seq_id(
        (vseq.m) - (vseq.n)))).0;
        thus rseq is convergent & lim rseq = Partial_Sums(abs(seq_id tseq-
        seq_id(vseq.n))).0
        proof
          consider rseq0 be Real_Sequence such that
A31:      for m be Nat holds rseq0.m=(seq_id(vseq.m)).0 and
A32:      rseq0 is convergent and
A33:      tseq.0=lim rseq0 by A11;
A34:      for m being Nat holds rseq.m = |.rseq0.m-(seq_id((
          vseq.n))).0 .|
          proof
            let m be Nat;
            rseq.m=Partial_Sums(abs(seq_id((vseq.m) - (vseq.n)))).0 by A30
              .=(abs(seq_id((vseq.m) - (vseq.n)))).0 by SERIES_1:def 1
              .=(abs(seq_id(vseq.m)-seq_id(vseq.n))).0 by A20
              .=(abs(seq_id(vseq.m)+-seq_id(vseq.n))).0 by SEQ_1:11
              .=|.(seq_id(vseq.m)+-seq_id(vseq.n)).0 .| by SEQ_1:12
              .=|.(seq_id(vseq.m)).0+(-seq_id(vseq.n)).0 .| by SEQ_1:7
              .=|.(seq_id(vseq.m)).0+-(seq_id(vseq.n)).0 .| by SEQ_1:10
              .=|.(seq_id(vseq.m)).0-(seq_id(vseq.n)).0 .|;
            hence thesis by A31;
          end;
          then lim rseq = |.lim(rseq0) -((seq_id(vseq.n)).0 ).| by A32,Th1
            .= |.tseq.0+-((seq_id(vseq.n)).0).| by A33
            .= |.tseq.0+(-(seq_id(vseq.n))).0 .| by SEQ_1:10
            .= |.(tseq+(-seq_id((vseq.n)))).0 .| by SEQ_1:7
            .= |.(tseq-(seq_id((vseq.n)))).0 .| by SEQ_1:11
            .= abs(seq_id tseq-(seq_id(vseq.n))).0 by SEQ_1:12
            .=Partial_Sums(abs(seq_id tseq-(seq_id(vseq.n)))).0 by
SERIES_1:def 1;
          hence thesis by A32,A34,Th1;
        end;
      end;
      then
A35:  P[0];
      for i be Nat holds P[i] from NAT_1:sch 2(A35,A29);
      hence thesis;
    end;
    for n be Nat st n >= k holds abs(seq_id tseq-seq_id(vseq.n
    )) is summable & Sum abs(seq_id tseq-seq_id(vseq.n)) < e1
    proof
      let n be Nat such that
A36:  n >= k;
A37:  for i be Nat st 0 <= i holds Partial_Sums(abs((seq_id
      tseq -seq_id(vseq.n)))).i <=e
      proof
        let i be Nat such that
        0 <=i;
        deffunc F(Nat)= Partial_Sums(abs(seq_id((vseq.$1) - (vseq.n
        )))).i;
        consider rseq be Real_Sequence such that
A38:    for m be Nat holds rseq.m = F(m) from SEQ_1:sch 1;
A39:    for m be Nat st m >= k holds rseq.m <= e
        proof
          let m be Nat;
A40:      rseq.m = Partial_Sums(abs(seq_id((vseq.m) - (vseq.n)))).i by A38;
          assume
A41:      m >= k;
          then
          abs(seq_id((vseq.m) - (vseq.n))) is summable &
          for i be Nat holds 0 <= abs(seq_id((vseq.m) - (vseq.n))).i
                 by A16,A36;
          then
A42:      Partial_Sums(abs(seq_id((vseq.m) - (vseq.n)))).i <=Sum(abs(
          seq_id((vseq.m) - (vseq.n)))) by RSSPACE2:3;
          Sum(abs(seq_id((vseq.m) - (vseq.n)))) < e by A16,A36,A41;
          hence thesis by A42,A40,XXREAL_0:2;
        end;
        rseq is convergent & lim rseq = Partial_Sums(abs(seq_id tseq-
        seq_id(vseq.n)) ).i by A19,A38;
        hence thesis by A39,RSSPACE2:5;
      end;
      now
        take e1;
        let i be Nat;
        Partial_Sums(abs((seq_id tseq -seq_id(vseq.n)))).i <=e by A37,NAT_1:2;
        hence Partial_Sums(abs((seq_id tseq -seq_id(vseq.n)))).i < e1 by A14,
XXREAL_0:2;
      end;
      then
A43:  Partial_Sums(abs((seq_id tseq -seq_id(vseq.n)))) is bounded_above
      by SEQ_2:def 3;
A44:  for i be Nat holds 0 <= abs(seq_id tseq-seq_id(vseq.n)). i
      proof
        let i be Nat;
        abs(seq_id tseq -seq_id(vseq.n)).i =|.(seq_id tseq -seq_id(vseq
        .n)).i.| by SEQ_1:12;
        hence thesis by COMPLEX1:46;
      end;
      then abs((seq_id tseq-seq_id(vseq.n))) is summable by A43,SERIES_1:17;
      then Partial_Sums(abs((seq_id tseq -seq_id(vseq.n)))) is convergent by
SERIES_1:def 2;
      then Sum(abs((seq_id tseq -seq_id(vseq.n)))) = lim Partial_Sums(abs((
seq_id tseq - seq_id(vseq.n)))) & lim Partial_Sums(abs((seq_id tseq -seq_id(
      vseq.n)))) <= e by A37,RSSPACE2:5,SERIES_1:def 3;
      hence thesis by A14,A44,A43,SERIES_1:17,XXREAL_0:2;
    end;
    hence thesis;
  end;
  abs seq_id tseq is summable
  proof
    set d=abs seq_id tseq;
A45: for i be Nat holds 0 <= abs(seq_id tseq).i
    proof
      let i be Nat;
      abs(seq_id tseq).i = |.(seq_id tseq).i.| by SEQ_1:12;
      hence thesis by COMPLEX1:46;
    end;
    consider m be Nat such that
A46: for n be Nat st n >= m holds abs((seq_id tseq -seq_id(
    vseq.n))) is summable & Sum(abs((seq_id tseq -seq_id(vseq.n)))) < 1 by A12;
    set b=abs seq_id(vseq.m);
    set a=abs(seq_id tseq -seq_id(vseq.m));
    seq_id(vseq.m) is absolutely_summable by Def1;
    then
A47: abs(seq_id(vseq.m)) is summable by SERIES_1:def 4;
A48: for i be Nat holds d.i <= (a+b).i
    proof
      let i be Nat;
A49:  b.i=|.(seq_id(vseq.m)).i.| & d.i=|.(seq_id tseq).i.| by SEQ_1:12;
      a.i = |.(seq_id tseq -seq_id(vseq.m)).i.| by SEQ_1:12
        .= |.(seq_id tseq+-seq_id(vseq.m)).i.| by SEQ_1:11
        .= |.(seq_id tseq).i+(-seq_id(vseq.m)).i.| by SEQ_1:7
        .= |.(seq_id tseq).i+(-(seq_id(vseq.m)).i).| by SEQ_1:10
        .=|.(seq_id tseq).i-(seq_id(vseq.m)).i.|;
      then d.i-b.i <= a.i by A49,COMPLEX1:59;
      then d.i-b.i+b.i<= a.i + b.i by XREAL_1:6;
      hence thesis by SEQ_1:7;
    end;
    abs((seq_id tseq -seq_id(vseq.m))) is summable by A46;
    then a + b is summable by A47,SERIES_1:7;
    hence thesis by A45,A48,SERIES_1:20;
  end;
  then
A50: seq_id tseq is absolutely_summable by SERIES_1:def 4;
A51: tseq in the_set_of_RealSequences by FUNCT_2:8;
  then reconsider tv=tseq as Point of l1_Space by A50,Def1;
  for e be Real st e > 0
    ex m be Nat st
   for n be Nat st n >= m holds ||.(vseq.n) - tv.|| < e
  proof
    let e be Real;
    assume e > 0;
    then consider m be Nat such that
A52: for n be Nat st n >= m holds abs(seq_id tseq-seq_id(
    vseq.n)) is summable & Sum(abs(seq_id tseq-seq_id(vseq.n))) < e by A12;
    reconsider m as Element of NAT by ORDINAL1:def 12;
    take m;
      reconsider u=tseq as VECTOR of l1_Space by A50,A51,Def1;
      let n be Nat;
      assume n >= m;
      then
A53:  Sum abs(seq_id tseq-seq_id(vseq.n)) < e by A52;
      reconsider v=vseq.n as VECTOR of l1_Space;
      seq_id(u-v) = u-v by Th6;
      then Sum abs seq_id(u-v) = Sum abs(seq_id tseq-seq_id(vseq.n)) by Th6;
      then
A54:  (the normF of l1_Space).(u-v) < e by A53,Def2;
      ||.(vseq.n) - tv.|| =||.-(tv-(vseq.n)).|| by RLVECT_1:33
        .=||.tv-(vseq.n).|| by NORMSP_1:2;
      hence ||.(vseq.n) - tv.|| < e by A54;
  end;
  hence thesis by NORMSP_1:def 6;
end;
