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 Th21:
  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 ExpSeq).l * (w ExpSeq).(((k+1)-' l)) =1r/(l!)*(z #N l) * (w ExpSeq).
  (((k+1)-' l)) by Def1
    .=1r/(l!)*(z #N l) * (1r/(((k+1)-'l )!)*(w #N (((k+1)-'l)))) by Def1
    .=(1r/(l!)* (1r/(((k+1)-'l)!))*((z #N l) *(w #N (((k+1)-'l))))) by
CLOPBAN3:38
    .=((1r*1r)/((l!) * (((k+1)-'l)!) )*((z #N l) *(w #N (((k+1)-'l))))) by
XCMPLX_1:76
    .= ((Coef_e(k+1)).l) * ((z #N l) * (w #N (((k+1)-'l)))) by A3,
COMPLEX1:def 4,SIN_COS:def 7
    .= ((Coef_e(k+1)).l) * (z #N l) * (w #N (((k+1)-'l))) by CLOPBAN3:38
    .=Expan_e(k+1,z,w).l by A3,Def3;
  (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 ExpSeq).l )* (Partial_Sums(w ExpSeq).((k -' l)+1)
  ) by A3,Def4
    .=(z ExpSeq).l * (Partial_Sums(w ExpSeq).((k -' l)) +(w ExpSeq).(((k+1)
  -'l))) by A5,BHSP_4:def 1
    .=((z ExpSeq).l * (Partial_Sums(w ExpSeq).((k -' l))) +((z ExpSeq).l * (
  w ExpSeq).(( (k+1) -' l)))) by VECTSP_1:def 2
    .=(Alfa(k,z,w)).l+((z ExpSeq).l * (w ExpSeq).(( (k+1) -' l))) by A1,Def4;
  hence thesis by A4;
end;
