reserve X for Complex_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 Th24:
  for z,w st z,w are_commutative holds Partial_Sums((z+w) ExpSeq).
  n = Partial_Sums(Alfa(n,z,w)).n
proof
  let z,w such that
A1: z,w are_commutative;
  defpred X[Nat] means
Partial_Sums((z+w) ExpSeq).$1=Partial_Sums(
  Alfa($1,z,w)).$1;
A2: for k st X[k] holds X[k+1]
  proof
    let k such that
A3: Partial_Sums((z+w) ExpSeq).k=Partial_Sums(Alfa(k,z,w)).k;
    k+1-'(k+1)=0 by XREAL_1:232;
    then
    (Alfa(k+1,z,w)).(k+1) =(z ExpSeq).(k+1) * Partial_Sums(w ExpSeq).0 by Def4
      .=(z ExpSeq).(k+1) * ((w ExpSeq).0) by BHSP_4:def 1
      .=(z ExpSeq).(k+1) * 1.X by Th20
      .=(z ExpSeq).(k+1) by VECTSP_1:def 4
      .=(Expan_e(k+1,z,w)).(k+1) by Th23;
    then
A4: (Partial_Sums(( Expan_e(k+1,z,w) ))).k + (Alfa(k+1,z,w)).(k+1) =(
    Partial_Sums(( Expan_e(k+1,z,w) ))).(k+1) by BHSP_4:def 1
      .=1r/((k+1)!) * (z+w) #N (k+1) by A1,Th18;
    Partial_Sums((Alfa(k+1,z,w))).(k+1) =Partial_Sums((Alfa(k+1,z,w))).k+(
    Alfa(k+1,z,w)).(k+1) by BHSP_4:def 1
      .=((Partial_Sums(Alfa(k,z,w))).k + (Partial_Sums(Expan_e(k+1,z,w))).k)
    + (Alfa(k+1,z,w)).(k+1) by Th22
      .= Partial_Sums((z+w) ExpSeq).k + ((Partial_Sums(Expan_e(k+1,z,w))).k
    + (Alfa(k+1,z,w)).(k+1)) by A3,RLVECT_1:def 3;
    then Partial_Sums((Alfa(k+1,z,w))).(k+1) = Partial_Sums((z+w) ExpSeq).k +
    (z+w) ExpSeq.(k+1) by A4,Def1
      .= Partial_Sums((z+w) ExpSeq).(k+1) by BHSP_4:def 1;
    hence Partial_Sums((z+w) ExpSeq).(k+1)=Partial_Sums(Alfa(k+1,z,w)).(k+1);
  end;
A5: Partial_Sums((z+w) ExpSeq).0 =((z+w) ExpSeq).0 by BHSP_4:def 1
    .=1.X by Th20;
A6: 0-'0=0 by XREAL_1:232;
  Partial_Sums(Alfa(0,z,w)).0 = Alfa(0,z,w).0 by BHSP_4:def 1
    .= (z ExpSeq).0 * Partial_Sums(w ExpSeq).0 by A6,Def4
    .= (z ExpSeq).0 * (w ExpSeq).0 by BHSP_4:def 1
    .=1.X * (w ExpSeq).0 by Th20
    .=1.X * 1.X by Th20
    .= 1.X by VECTSP_1:def 4;
  then
A7: X[0] by A5;
  for n holds X[n] from NAT_1:sch 2(A7,A2);
  hence thesis;
end;
