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 Th12:
  Partial_Sums((Alfa(k+1,z,w))).k = (Partial_Sums(( Alfa(k,z,w)))).k
  + (Partial_Sums(( Expan_e(k+1,z,w) ))).k
proof
A1: k in NAT by ORDINAL1:def 12;
 now
    let l be Nat;
A2: l in NAT by ORDINAL1:def 12;
    assume l <= k;
    hence
(Alfa(k+1,z,w)).l = (Alfa(k,z,w)).l + Expan_e(k+1,z,w).l by Th11
      .= ( (Alfa(k,z,w)) + Expan_e(k+1,z,w)).l by VALUED_1:1,A2;
  end;
  hence Partial_Sums((Alfa(k+1,z,w))).k
  =Partial_Sums(((Alfa(k,z,w)) + Expan_e(k+1,z,w))).k by COMSEQ_3:35
    .=(Partial_Sums(( Alfa(k,z,w)))
  +Partial_Sums(( Expan_e(k+1,z,w )))).k by COMSEQ_3:27
    .= (Partial_Sums(( Alfa(k,z,w)))).k
  + (Partial_Sums(( Expan_e(k+1,z,w) ))).k by VALUED_1:1,A1;
end;
