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 Th28:
 for k being Nat holds
  ||. Partial_Sums((z rExpSeq)).k .|| <= Partial_Sums(||.z.||
  rExpSeq).k & Partial_Sums((||.z.|| rExpSeq)).k <= Sum(||.z.|| rExpSeq) & ||.
  Partial_Sums(( z rExpSeq)).k .|| <= Sum(||.z.|| rExpSeq)
proof let k be Nat;
  defpred X[Nat] means
||. Partial_Sums(( z rExpSeq)).$1 .|| <=
  Partial_Sums((||.z.|| rExpSeq)).$1;
A1: Partial_Sums((||.z.|| rExpSeq)).0 = ((||.z.|| rExpSeq)).0 by SERIES_1:def 1
    .= ((||.z.|| ) |^ 0 ) / (0!) by SIN_COS:def 5
    .= 1 by NEWTON:4,12;
A2: for k being Nat st X[k] holds X[k+1]
  proof
    let k be Nat;
    assume
    ||. Partial_Sums(( z rExpSeq)).k .|| <= Partial_Sums((||.z.|| rExpSeq)).k;
    then
A3: ||. Partial_Sums(( z rExpSeq)).k.|| + (||.z.|| rExpSeq).(k+1) <=
    Partial_Sums((||.z.|| rExpSeq)).k + (||.z.|| rExpSeq).(k+1) by XREAL_1:6;
A4: ||. Partial_Sums(( z rExpSeq)).k + ((z rExpSeq)).(k+1) .|| <= ||.
Partial_Sums(( z rExpSeq)).k.|| + ||. ((z rExpSeq)).(k+1) .|| by LOPBAN_3:38;
    ||. ((z rExpSeq)).(k+1) .|| <=(||.z.|| rExpSeq).(k+1) by Th14;
    then
A5: ||. Partial_Sums(( z rExpSeq)).k.|| + ||. ((z rExpSeq)).(k+1) .|| <=
    ||. Partial_Sums(( z rExpSeq)).k.|| + (||.z.|| rExpSeq).(k+1) by XREAL_1:7;
A6: Partial_Sums((||.z.|| rExpSeq)).k + (||.z.|| rExpSeq).(k+1) =
    Partial_Sums((||.z.|| rExpSeq)).(k+1) by SERIES_1:def 1;
    ||. Partial_Sums(( z rExpSeq)).(k+1) .|| =||. Partial_Sums(( z rExpSeq
    )).k + ((z rExpSeq)).(k+1) .|| by BHSP_4:def 1;
    then ||. Partial_Sums(( z rExpSeq)).(k+1) .|| <= ||. Partial_Sums(( z
    rExpSeq)).k.|| + (||.z.|| rExpSeq).(k+1) by A4,A5,XXREAL_0:2;
    hence thesis by A3,A6,XXREAL_0:2;
  end;
  ||. Partial_Sums(( z rExpSeq)).0 .|| = ||. ((z rExpSeq)).0 .|| by
BHSP_4:def 1
    .= ||. 1/ (0! )*(z #N 0).|| by Def2
    .= ||. 1 /1 *1.X.|| by LOPBAN_3:def 9,NEWTON:12
    .= ||. 1.X.|| by LOPBAN_3:38
    .= 1 by LOPBAN_3:38;
  then
A7: X[0] by A1;
A8: for k being Nat holds X[k] from NAT_1:sch 2(A7,A2);
  hence
  ||. Partial_Sums(( z rExpSeq)).k .|| <= Partial_Sums((||.z.|| rExpSeq)) .k;
A9: for n being Nat holds 0 <= (||.z.|| rExpSeq).n by Th27;
  ||.z.|| rExpSeq is summable by SIN_COS:45;
  then
A10: Partial_Sums(||.z.|| rExpSeq) is bounded_above by A9,SERIES_1:17;
  then Partial_Sums((||.z.|| rExpSeq)).k <= lim(Partial_Sums((||.z.|| rExpSeq
  ))) by A9,SEQ_4:37,SERIES_1:16;
  hence Partial_Sums((||.z.|| rExpSeq)).k <= Sum(||.z.|| rExpSeq) by
SERIES_1:def 3;
  now
    let k be Nat;
    lim(Partial_Sums((||.z.|| rExpSeq)))=Sum(||.z.|| rExpSeq) by SERIES_1:def 3
;
    hence
    Partial_Sums((||.z.|| rExpSeq)).k <= Sum(||.z.|| rExpSeq) by A9,A10,
SEQ_4:37,SERIES_1:16;
  end;
  then
A11: Partial_Sums((||.z.|| rExpSeq)).k <= Sum(||.z.|| rExpSeq);
  ||. Partial_Sums(( z rExpSeq)).k .|| <= Partial_Sums((||.z.|| rExpSeq))
  .k by A8;
  hence thesis by A11,XXREAL_0:2;
end;
