reserve X for Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th32:
  for p being Real st p>0 ex n st for k st n <= k holds
  |.Partial_Sums(||.Conj(k,z,w).||).k.| < p
proof
  let p be Real such that
A1: p>0;
  reconsider pp = p as Real;
  set p1=min((pp/(4 * Sum(||.z.|| rExpSeq))),(pp/(4 * Sum(||.w.|| rExpSeq))));
A2: 1 <= Sum(||.z.|| rExpSeq) by Th29;
  then 0 * Sum(||.z.|| rExpSeq) < 4 * Sum(||.z.|| rExpSeq) by XREAL_1:68;
  then
A3: 0/(4 * Sum(||.z.|| rExpSeq)) < p/(4 * Sum(||.z.|| rExpSeq)) by A1,
XREAL_1:74;
A4: 1 <= Sum(||.w.|| rExpSeq) by Th29;
  then 0 * Sum(||.w.|| rExpSeq) < 4 * Sum(||.w.|| rExpSeq) by XREAL_1:68;
  then 0/(4 * Sum(||.w.|| rExpSeq)) < p/(4 * Sum(||.w.|| rExpSeq)) by A1,
XREAL_1:74;
  then
A5: p1 > 0 by A3,XXREAL_0:15;
  Partial_Sums(( w rExpSeq )) is convergent by LOPBAN_3:def 1;
  then for s be Real st 0<s ex n be Nat st for m be Nat
  st n<=m holds ||.Partial_Sums(( w rExpSeq )).m -Partial_Sums(( w rExpSeq )).n
  .||<s by LOPBAN_3:4;
  then Partial_Sums(( w rExpSeq )) is Cauchy_sequence_by_Norm by LOPBAN_3:5;
  then consider n2 such that
A6: for k,l be Nat st n2 <= k & n2 <= l holds ||.
  Partial_Sums(( w rExpSeq )).k - Partial_Sums(( w rExpSeq )).l.|| < p1 by A5,
RSSPACE3:8;
  ( ||.z.|| rExpSeq ) is summable by SIN_COS:45;
  then Partial_Sums(( ||.z.|| rExpSeq )) is convergent by SERIES_1:def 2;
  then consider n1 being Nat such that
A7: for k,l be Nat st n1 <= k & n1 <= l holds |.
Partial_Sums(( ||.z.|| rExpSeq )).k - Partial_Sums(( ||.z.|| rExpSeq )).l.|
     < p1
  by A5,COMSEQ_3:4;
  set n3=n1+n2;
  take n4 = n3+n3;
A8: now
    let n;
    let k be Nat;
      let l be Nat such that
A9:   l <= k;
A10:  ||.(z rExpSeq).l * (Partial_Sums(w rExpSeq).k -Partial_Sums(w
rExpSeq).(k-'l)) .|| <=||.(z rExpSeq).l.|| * ||.(Partial_Sums(w rExpSeq).k -
      Partial_Sums(w rExpSeq).(k-'l)).|| by LOPBAN_3:38;
      0 <=||.(Partial_Sums(w rExpSeq).k -Partial_Sums(w rExpSeq).(k-'l))
      .|| by NORMSP_1:4;
      then
A11:  ||.(z rExpSeq).l.|| * ||.(Partial_Sums(w rExpSeq).k -Partial_Sums(w
rExpSeq).(k-'l)).|| <=(||.z.|| rExpSeq).l * ||.(Partial_Sums(w rExpSeq).k -
      Partial_Sums(w rExpSeq).(k-'l)).|| by Th14,XREAL_1:64;
      ||.Conj(k,z,w).||.l =||.( Conj(k,z,w).l) .|| by NORMSP_0:def 4
        .=||.(z rExpSeq).l * (Partial_Sums(w rExpSeq).k -Partial_Sums(w
      rExpSeq).(k-'l)) .|| by A9,Def9;
      hence ||.Conj(k,z,w).||.l <=(||.z.|| rExpSeq).l * ||.(Partial_Sums(w
      rExpSeq).k -Partial_Sums(w rExpSeq).(k-'l)).|| by A10,A11,XXREAL_0:2;
  end;
A12: now
    let k be Nat;
      let l be Nat;
A13:  ||.Partial_Sums(w rExpSeq).k -Partial_Sums(w rExpSeq).(k-'l).|| <=
||.Partial_Sums(w rExpSeq).k .|| + ||.Partial_Sums(w rExpSeq).(k-'l).|| by
NORMSP_1:3;
      ||.Partial_Sums(w rExpSeq).(k-'l).|| <= Sum(||.w.|| rExpSeq) by Th28;
      then
A14:  Sum(||.w.|| rExpSeq) + ||.Partial_Sums(w rExpSeq).(k-'l).|| <= Sum(
      ||.w.|| rExpSeq)+ Sum(||.w.|| rExpSeq) by XREAL_1:6;
      ||.Partial_Sums(w rExpSeq).k .|| <= Sum(||.w.|| rExpSeq) by Th28;
      then
      ||.Partial_Sums(w rExpSeq).k .|| + ||.Partial_Sums(w rExpSeq).(k -'
      l).|| <= Sum(||.w.|| rExpSeq) + ||.Partial_Sums(w rExpSeq).(k-'l).|| by
XREAL_1:6;
      then
      ||.Partial_Sums(w rExpSeq).k .|| + ||.Partial_Sums(w rExpSeq).(k-'l
      ).|| <= 2 * Sum(||.w.|| rExpSeq) by A14,XXREAL_0:2;
      then
A15:  ||.Partial_Sums(w rExpSeq).k -Partial_Sums(w rExpSeq).(k-'l).|| <=
      2 * Sum(||.w.|| rExpSeq) by A13,XXREAL_0:2;
      assume l <= k;
      then
A16:  ||.Conj(k,z,w).||.l <= (||.z.|| rExpSeq).l * ||.Partial_Sums(w
      rExpSeq).k -Partial_Sums(w rExpSeq).(k-'l).|| by A8;
      0 <= (||.z.|| rExpSeq).l by Th27;
      then (||.z.|| rExpSeq).l * ||.Partial_Sums(w rExpSeq).k -Partial_Sums(w
rExpSeq).(k-'l).|| <= (||.z.|| rExpSeq).l * (2 * Sum(||.w.|| rExpSeq)) by A15,
XREAL_1:64;
      hence ||.Conj(k,z,w).||.l <= (||.z.|| rExpSeq).l * (2 * Sum(||.w.||
      rExpSeq)) by A16,XXREAL_0:2;
  end;
  now
    0 < p/4 by A1,XREAL_1:224;
    then
A17: 0+p/4 < p/4 + p/4 by XREAL_1:6;
A18: 0 <> Sum(||.z.|| rExpSeq) by Th29;
A19: Sum(||.z.|| rExpSeq) * (p/(4 * Sum(||.z.|| rExpSeq))) =Sum(||.z.||
    rExpSeq) * (p * (4 * Sum(||.z.|| rExpSeq))") by XCMPLX_0:def 9
      .=(Sum(||.z.|| rExpSeq) * p) * (4 * Sum(||.z.|| rExpSeq))"
      .=(Sum(||.z.|| rExpSeq) * p)/(4 * Sum(||.z.|| rExpSeq)) by XCMPLX_0:def 9
      .=p/4 by A18,XCMPLX_1:91;
    let k such that
A20: n4 <= k;
A21: 0+n3 <= n3+n3 by XREAL_1:6;
    then k-'n3=k-n3 by A20,XREAL_1:233,XXREAL_0:2;
    then
A22: k=(k-'n3)+n3;
A23: n3 <= k by A20,A21,XXREAL_0:2;
    now
      let l be Nat;
A24:  0+n2 <= n1+n2 by XREAL_1:6;
      0+n3 <= n3+n3 by XREAL_1:6;
      then n2 <= n4 by A24,XXREAL_0:2;
      then
A25:  n2 <= k by A20,XXREAL_0:2;
A26:  0 <= (||.z.|| rExpSeq).l by Th27;
      assume
A27:  l <= n3;
      then
A28:  n3+n3-n3 <= n3+n3-l by XREAL_1:10;
      l <= k by A23,A27,XXREAL_0:2;
      then
A29:  ||.Conj(k,z,w).||.l <=(||.z.|| rExpSeq).l * ||.(Partial_Sums(w
      rExpSeq).k -Partial_Sums(w rExpSeq).(k-'l)).|| by A8;
      n4-l <= k-l by A20,XREAL_1:9;
      then
A30:  n3 <= k-l by A28,XXREAL_0:2;
      k-'l=k-l by A23,A27,XREAL_1:233,XXREAL_0:2;
      then n2 <= k-'l by A24,A30,XXREAL_0:2;
      then ||.Partial_Sums(( w rExpSeq )).k - Partial_Sums(( w rExpSeq )).( k
      -'l).|| < p1 by A6,A25;
      then (||.z.|| rExpSeq).l * ||.Partial_Sums(( w rExpSeq )).k -
      Partial_Sums(( w rExpSeq )).(k-'l).|| <= (||.z.|| rExpSeq).l * p1 by A26,
XREAL_1:64;
      hence ||.Conj(k,z,w).||.l <= p1 * (||.z.|| rExpSeq).l by A29,XXREAL_0:2;
    end;
    then
A31: Partial_Sums(||.Conj(k,z,w).||).n3 <= Partial_Sums(||.z.|| rExpSeq).
    n3 * p1 by COMSEQ_3:7;
A32: n1+0 <= n1+n2 by XREAL_1:6;
    then n1 <= k by A23,XXREAL_0:2;
    then
    |.(Partial_Sums(||.z.|| rExpSeq).k-Partial_Sums(||.z.|| rExpSeq).n3
    ) .| <= p1 by A7,A32;
    then Partial_Sums(||.z.|| rExpSeq).k-Partial_Sums(||.z.|| rExpSeq).n3 <=
    p1 by A20,A21,Th30,XXREAL_0:2;
    then
A33: (Partial_Sums(||.z.|| rExpSeq).k -Partial_Sums(||.z.|| rExpSeq).n3) *
    (2 * Sum(||.w.|| rExpSeq)) <= p1 * (2 * Sum(||.w.|| rExpSeq)) by A4,
XREAL_1:64;
    for l be Nat st l <= k holds ||.Conj(k,z,w).||.l <= (2 *
    Sum(||.w.|| rExpSeq)) * (||.z.|| rExpSeq).l by A12;
    then
    Partial_Sums(||.Conj(k,z,w).||).(k) -Partial_Sums(||.Conj(k,z,w) .||)
.n3 <= (Partial_Sums(||.z.|| rExpSeq).(k) -Partial_Sums(||.z.|| rExpSeq).n3) *(
    2 * Sum(||.w.|| rExpSeq)) by A22,COMSEQ_3:8;
    then
A34: Partial_Sums(||.Conj(k,z,w).||).k-Partial_Sums(||.Conj(k,z,w).||) .n3
    <= p1 * (2 * Sum(||.w.|| rExpSeq)) by A33,XXREAL_0:2;
A35: 0 <> Sum(||.w.|| rExpSeq) by Th29;
    Partial_Sums(||.z.|| rExpSeq).n3 * p1 <= Sum(||.z.|| rExpSeq) * p1 by A5
,Th28,XREAL_1:64;
    then
A36: Partial_Sums(||.Conj(k,z,w).||).n3 <= Sum(||.z.|| rExpSeq) * p1 by A31,
XXREAL_0:2;
    Sum(||.z.|| rExpSeq) * p1 <= Sum(||.z.|| rExpSeq) * (p/(4 * Sum(||.z
    .|| rExpSeq))) by A2,XREAL_1:64,XXREAL_0:17;
    then Partial_Sums(||.Conj(k,z,w).||).n3 <= p/4 by A36,A19,XXREAL_0:2;
    then
A37: Partial_Sums(||.Conj(k,z,w).||).n3 < p/2 by A17,XXREAL_0:2;
    0 * Sum(||.w.|| rExpSeq) < 2 * Sum(||.w.|| rExpSeq) by A4,XREAL_1:68;
    then
A38: (2 * Sum(||.w.|| rExpSeq)) * p1 <= (2 * Sum(||.w.|| rExpSeq)) * (p/(4
    * Sum(||.w.|| rExpSeq))) by XREAL_1:64,XXREAL_0:17;
    (2 * Sum(||.w.|| rExpSeq) ) * (p/(4 * Sum(||.w.|| rExpSeq))) =(2 *
Sum(||.w.|| rExpSeq) ) * (p * (4 * Sum (||.w.|| rExpSeq))") by XCMPLX_0:def 9
      .=((2 * Sum(||.w.|| rExpSeq) ) * p ) * (4 * Sum (||.w.|| rExpSeq))"
      .=((2 * p ) * Sum(||.w.|| rExpSeq) ) /(4 * Sum(||.w.|| rExpSeq)) by
XCMPLX_0:def 9
      .= (2 * p )/4 by A35,XCMPLX_1:91
      .= p/2;
    then
    Partial_Sums(||.Conj(k,z,w).||).k -Partial_Sums(||.Conj(k,z,w).||).n3
    <= p/2 by A34,A38,XXREAL_0:2;
    then (Partial_Sums(||.Conj(k,z,w).||).k -Partial_Sums(||.Conj(k,z,w).||).
    n3) +Partial_Sums(||.Conj(k,z,w).||).n3 < (p/2)+(p/2) by A37,XREAL_1:8;
    hence |.Partial_Sums(||.Conj(k,z,w).||).k.| < p by Th31;
  end;
  hence thesis;
end;
