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 Th25:
  for z,w st z,w are_commutative holds Partial_Sums((z+w) rExpSeq)
  .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) rExpSeq).$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) rExpSeq).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 rExpSeq).(k+1) * Partial_Sums(w rExpSeq).0
    by Def8
      .=(z rExpSeq).(k+1) * ((w rExpSeq).0) by BHSP_4:def 1
      .=(z rExpSeq).(k+1) * 1.X by Th21
      .=(z rExpSeq).(k+1) by LOPBAN_3:38
      .=(Expan_e(k+1,z,w)).(k+1) by Th24;
    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
      .=1/((k+1)!)* (z+w) #N (k+1) by A1,Th19;
    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 Th23
      .= Partial_Sums((z+w) rExpSeq).k + ((Partial_Sums(( Expan_e(k+1,z,w) )
    )).k + (Alfa(k+1,z,w)).(k+1)) by A3,LOPBAN_3:38;
    then
    Partial_Sums((Alfa(k+1,z,w))).(k+1) =Partial_Sums((z+w) rExpSeq).k +(
    z+w) rExpSeq.(k+1) by A4,Def2
      .=Partial_Sums((z+w) rExpSeq).(k+1) by BHSP_4:def 1;
    hence Partial_Sums((z+w) rExpSeq).(k+1)=Partial_Sums(Alfa(k+1,z,w)).(k+1);
  end;
A5: Partial_Sums((z+w) rExpSeq).0 =((z+w) rExpSeq).0 by BHSP_4:def 1
    .=1.X by Th21;
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 rExpSeq).0 * Partial_Sums(w rExpSeq).0 by A6,Def8
    .= (z rExpSeq).0 * (w rExpSeq).0 by BHSP_4:def 1
    .=1.X * (w rExpSeq).0 by Th21
    .=1.X * 1.X by Th21
    .= 1.X by LOPBAN_3:38;
  then
A7: X[0] by A5;
  for n holds X[n] from NAT_1:sch 2(A7,A2);
  hence thesis;
end;
