reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;

theorem Th11:
  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;
 (k+1-'l)=k+1-l by A1,A2,XREAL_1:233,XXREAL_0:2;
then A4: (k+1-'l)=k-l+1
    .=(k-'l)+1 by A1,XREAL_1:233;
then A5: (Alfa(k+1,z,w)).l
  =(z ExpSeq).l * (Partial_Sums(w ExpSeq).((k-'l)+1)) by A3,Def11
    .=(z ExpSeq).l * (Partial_Sums(w ExpSeq).((k-'l))
  +(w ExpSeq).((k+1-'l))) by A4,SERIES_1:def 1
    .=((z ExpSeq).l * (Partial_Sums(w ExpSeq).((k-'l)))
  +((z ExpSeq).l * (w ExpSeq).((k+1-'l))))
    .=(Alfa(k,z,w)).l+((z ExpSeq).l * (w ExpSeq).((k+1-'l))) by A1,Def11;
 (z ExpSeq).l * (w ExpSeq).((k+1-'l))
  =(z |^ l/(l! )) * (w ExpSeq).((k+1-'l)) by Def4
    .=(z |^ l/(l! )) * (w |^ ((k+1-'l))/(((k+1-'l))! )) by Def4
    .=(((z |^ l) * (w |^ ((k+1-'l))) * 1r)/((l! ) * (((k+1-'l))! )))
  by XCMPLX_1:76
    .=((z |^ l) * (w |^ ((k+1-'l))) * ((1r/((l! ) * (((k+1-'l))! )))))
    .= ((Coef_e(k+1)).l) * ((z |^ l) * (w |^ ((k+1-'l)))) by A3,Def7
    .= ((Coef_e(k+1)).l) * (z |^ l) * (w |^ ((k+1-'l)))
    .=Expan_e(k+1,z,w).l by A3,Def10;
  hence thesis by A5;
end;
