
theorem
  for vseq be sequence of linfty_Space st vseq is
  Cauchy_sequence_by_Norm holds vseq is convergent
proof
  let vseq be sequence of linfty_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,RSSPACE3:8;
        take k;
        let m be Nat such that
A6:     k<=m;
        seq_id((vseq.m)-(vseq.k)) is bounded by Def1;
        then
A7:     |.seq_id((vseq.m) - (vseq.k)).|.i <=
upper_bound rng |.seq_id((vseq.m)
        - (vseq.k)).| by Lm2;
        seq_id((vseq.m) - (vseq.k)) =seq_id(seq_id(vseq.m)-seq_id(vseq.k)
        ) by Th2
          .= seq_id(vseq.m)+-seq_id(vseq.k);
        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
A8:     |.rseqi.m-rseqi.k.| = abs(seq_id((vseq.m) - (vseq.k))).i by SEQ_1:12;
        ||.(vseq.m) - (vseq.k).|| =
        upper_bound rng |.seq_id((vseq.m) - (vseq.k)
        ).| by Th2;
        then upper_bound rng |.seq_id((vseq.m) - (vseq.k)).| < e by A5,A6;
        hence thesis by A7,A8,XXREAL_0:2;
      end;
    end;
    then rseqi is convergent by SEQ_4:41;
    hence thesis by A3;
  end;
  consider f be sequence of REAL such that
A9: 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;
A10: 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 A9;
    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;
A11: 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 bounded &
upper_bound rng |.seq_id
  tseq-seq_id(vseq.n).| <= e
  proof
    let e be Real such that
A12: e > 0;
    reconsider e as Real;
    consider k be Nat such that
A13: for n, m be Nat st n >= k & m >= k holds ||.(vseq.n) -
    (vseq.m).|| < e by A1,A12,RSSPACE3:8;
A14: for m,n be Nat st n >= k & m >= k holds abs seq_id((vseq.n
    ) - (vseq.m)) is bounded &
    upper_bound rng abs seq_id((vseq.n) - (vseq.m)) < e
    proof
      let m,n be Nat;
      assume n >= k & m >= k;
      then
A15:  ||.(vseq.n) - (vseq.m).|| < e by A13;
      seq_id((vseq.n) - (vseq.m)) is bounded by Def1;
      hence thesis by A15,Def2;
    end;
A16: for n be Nat for i be Nat holds for rseq be
Real_Sequence st ( for m be Nat holds rseq.m=|.seq_id(vseq.m-vseq.
n).|.i ) holds rseq is convergent &
     lim rseq = abs((seq_id tseq -seq_id(vseq.n))
    ).i
    proof
      let n be Nat;
A17:  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 Th2;
        hence thesis;
      end;
      now
        let i be Nat;
        consider rseqi be Real_Sequence such that
A18:    for n be Nat holds rseqi.n=(seq_id(vseq.n)).i and
A19:    rseqi is convergent & tseq.i=lim rseqi by A10;
        now
          let rseq be Real_Sequence such that
A20:      for m be Nat holds rseq.m=abs(seq_id(vseq.m-vseq .n)).i;
A21:      now
            let m be Nat;
A22:        seq_id(vseq.m - vseq.n) = seq_id(vseq.m)-seq_id(vseq.n) by A17;
            thus rseq.m=abs(seq_id(vseq.m-vseq.n)).i by A20
              .=|.(seq_id(vseq.m-vseq.n)).i.| by SEQ_1:12
              .=|.(seq_id(vseq.m)).i -(seq_id(vseq.n)).i.| by A22,RFUNCT_2:1
              .=|.(rseqi.m) -(seq_id(vseq.n)).i.| by A18;
          end;
          |.tseq.i-(seq_id(vseq.n)).i.|
          = |.(tseq-(seq_id(vseq.n))).i.| by RFUNCT_2:1
          .= abs((seq_id tseq -seq_id(vseq.n))).i by SEQ_1:12;
          hence
          rseq is convergent & lim rseq = abs(seq_id tseq -seq_id(vseq.n)
          ).i by A19,A21,RSSPACE3:1;
        end;
        hence
        for rseq be Real_Sequence st ( for m be Nat holds rseq
.m=abs(seq_id(vseq.m-vseq.n)).i ) holds rseq is convergent & lim rseq = abs(
        seq_id tseq -seq_id(vseq.n)).i;
      end;
      hence thesis;
    end;
    for n be Nat st n >= k holds abs(seq_id tseq-seq_id(vseq.n
    )) is bounded & upper_bound rng |.seq_id tseq-seq_id(vseq.n).| <= e
    proof
      let n be Nat such that
A23:  n >= k;
A24:  for i be Nat holds abs((seq_id tseq -seq_id(vseq.n))).i <= e
      proof
        let i be Nat;
        deffunc F(Nat)= abs(seq_id((vseq.$1) - (vseq.n))).i;
        consider rseq be Real_Sequence such that
A25:    for m be Nat holds rseq.m = F(m) from SEQ_1:sch 1;
A26:    for m be Nat st m >= k holds rseq.m <= e
        proof
          let m be Nat;
A27:      rseq.m = abs(seq_id((vseq.m) - (vseq.n))).i by A25;
          assume
A28:      m >= k;
          then abs(seq_id((vseq.m) - (vseq.n))) is bounded by A14,A23;
          then
A29:      abs(seq_id((vseq.m) - (vseq.n))).i <=
upper_bound rng abs seq_id((vseq.m
          ) - (vseq.n)) by Lm2;
          upper_bound rng abs seq_id((vseq.m) - (vseq.n)) <= e by A14,A23,A28;
          hence thesis by A29,A27,XXREAL_0:2;
        end;
        rseq is convergent & lim rseq = abs(seq_id tseq-seq_id(vseq.n)).i
        by A16,A25;
        hence thesis by A26,RSSPACE2:5;
      end;
A30:  0 + e < 1 + e by XREAL_1:8;
      now
        let i be Nat;
        abs((seq_id tseq -seq_id(vseq.n))).i <= e & abs ((seq_id tseq -
seq_id(vseq.n ))).i =|.((seq_id tseq -seq_id(vseq.n))).i.| by A24,SEQ_1:12;
        hence |.((seq_id tseq -seq_id(vseq.n))).i.| <e+1 by A30,XXREAL_0:2;
      end;
      then seq_id tseq -seq_id(vseq.n) is bounded by A12,SEQ_2:3;
      hence thesis by A24,Lm1;
    end;
    hence thesis;
  end;
A31: seq_id tseq is bounded
  proof
    consider m be Nat such that
A32: for n be Nat st n >= m holds abs (seq_id tseq -seq_id(
vseq.n)) is bounded & upper_bound rng abs (seq_id tseq -seq_id(vseq.n))
 <= 1 by A11;
A33: abs (seq_id tseq -seq_id(vseq.m)) is bounded by A32;
    set d=abs seq_id tseq;
    set b=abs seq_id(vseq.m);
    set a=abs(seq_id tseq -seq_id(vseq.m));
A34: seq_id(vseq.m) is bounded by Def1;
A35: for i be Nat holds d.i <= (a+b).i
    proof
      let i be Nat;
A36:  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).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 A36,COMPLEX1:59;
      then d.i-b.i+b.i<= a.i + b.i by XREAL_1:6;
      hence thesis by SEQ_1:7;
    end;
    d is bounded
    proof
      reconsider r=upper_bound rng (a+b)+1 as Real;
      b.1=|. (seq_id(vseq.m)).1.| by SEQ_1:12;
      then
A37:  0<= b.1 by COMPLEX1:46;
A38:  upper_bound( rng(a+b) ) +0 < upper_bound( rng(a+b) )+1 by XREAL_1:8;
A39:  now
        let i be Nat;
        d.i <= (a+b).i & (a+b).i <= upper_bound rng (a+b) by A33,A34,A35,Lm2;
        then
A40:    d.i <= upper_bound rng (a+b) by XXREAL_0:2;
        d.i=|.(seq_id tseq).i.| by SEQ_1:12;
        hence |.(seq_id tseq).i.| <r by A38,A40,XXREAL_0:2;
      end;
      a.1=|.(seq_id tseq -seq_id(vseq.m)).1.| by SEQ_1:12;
      then (a+b).1 =a.1 + b.1 & 0<= a.1 by COMPLEX1:46,SEQ_1:7;
      then 0 <= upper_bound rng(a+b) by A33,A34,A37,Lm2;
      then seq_id tseq is bounded by A39,SEQ_2:3;
      hence thesis;
    end;
    hence thesis by SEQM_3:37;
  end;
A41: tseq in the_set_of_RealSequences by FUNCT_2:8;
  then reconsider tv=tseq as Point of linfty_Space by A31,Def1;
  take tv;
  let e1 be Real such that
A42: e1 > 0;
  set e=e1/2;
  consider m be Nat such that
A43: for n be Nat st n >= m holds |.seq_id tseq-seq_id(vseq
  .n).| is bounded & upper_bound rng abs(seq_id tseq-seq_id(vseq.n)) <= e
   by A11,A42,XREAL_1:215;
A44: e < e1 by A42,XREAL_1:216;
  now
    reconsider u=tseq as VECTOR of linfty_Space by A31,A41,Def1;
    let n be Nat;
    assume n >= m;
    then
A45: upper_bound rng( abs(seq_id tseq-seq_id(vseq.n))) <= e by A43;
    reconsider v=vseq.n as VECTOR of linfty_Space;
    seq_id(u-v) = u-v by Th2;
    then upper_bound rng abs seq_id(u-v) =
     upper_bound rng abs(seq_id tseq-seq_id(vseq.n)) by Th2;
    then
A46: (the normF of linfty_Space).(u-v) <= e by A45,Def2;
    ||.(vseq.n) - tv.|| =||.-(tv-(vseq.n)).|| by RLVECT_1:33
      .=||.tv-(vseq.n).|| by NORMSP_1:2;
    hence ||.(vseq.n) - tv.|| < e1 by A44,A46,XXREAL_0:2;
  end;
  hence thesis;
end;
