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 Th22:
  l <= k implies (Alfa(k+1,z,w)).l = (Alfa(k,z,w)).l + Expan_e(k+1 ,z,w).l
proof
  assume
A1: l <= k;
A2: k < k+1 by XREAL_1:29;
  then
A3: l <= k+1 by A1,XXREAL_0:2;
A4: (z rExpSeq).l * (w rExpSeq).(((k+1)-' l)) =1/(l! )*(z #N l) * (w rExpSeq
  ).(((k+1)-' l)) by Def2
    .=1/(l! )*(z #N l) * (1/(((k+1)-'l ) !)*(w #N (((k+1)-'l)))) by Def2
    .=(1/(l! )* (1/(((k+1)-'l ) !))*((z #N l) *(w #N (((k+1)-'l))))) by
LOPBAN_3:38
    .=(1/ ((l! )* (((k+1)-'l ) !))*((z #N l) *(w #N (((k+1)-'l))))) by
XCMPLX_1:102
    .= ((Coef_e(k+1)).l) * ((z #N l) * (w #N (((k+1)-'l)))) by A3,Def4
    .= ((Coef_e(k+1)).l) * (z #N l) * (w #N (((k+1)-'l))) by LOPBAN_3:38
    .=Expan_e(k+1,z,w).l by A3,Def7;
  (k+1-'l)=k+1-l by A1,A2,XREAL_1:233,XXREAL_0:2;
  then
A5: (k+1-'l)=k-l+1 .=(k-'l)+1 by A1,XREAL_1:233;
  then
  (Alfa(k+1,z,w)).l =((z rExpSeq).l )* (Partial_Sums(w rExpSeq).((k -' l)+
  1)) by A3,Def8
    .=(z rExpSeq).l * (Partial_Sums(w rExpSeq).((k -' l)) +(w rExpSeq).(((k+
  1)-'l))) by A5,BHSP_4:def 1
    .=((z rExpSeq).l * (Partial_Sums(w rExpSeq).((k -' l))) +((z rExpSeq).l
  * (w rExpSeq).(( (k+1) -' l)))) by LOPBAN_3:38
    .=(Alfa(k,z,w)).l+((z rExpSeq).l * (w rExpSeq).(( (k+1) -' l))) by A1,Def8;
  hence thesis by A4;
end;
