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 Th14:
  Partial_Sums((z+w) ExpSeq).n = Partial_Sums(Alfa(n,z,w)).n
proof
A1: Partial_Sums((z+w) ExpSeq).0 =((z+w) ExpSeq).0 by SERIES_1:def 1
    .=1 by Th10;
  defpred X[Nat] means
  Partial_Sums((z+w) ExpSeq).$1=Partial_Sums(Alfa($1,z,w)).$1;
A2: 0-'0=0 by XREAL_1:232;
 Partial_Sums(Alfa(0,z,w)).0 = Alfa(0,z,w).0 by SERIES_1:def 1
    .= (z ExpSeq).0 * Partial_Sums(w ExpSeq).0 by A2,Def11
    .= (z ExpSeq).0 * (w ExpSeq).0 by SERIES_1:def 1
    .=1r * (w ExpSeq).0 by Th10
    .= 1 by Th10;
then A3: X[0] by A1;
A4: for k st X[k] holds X[k+1]
  proof
    let k such that
A5: Partial_Sums((z+w) ExpSeq).k=Partial_Sums(Alfa(k,z,w)).k;
A6: 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 SERIES_1: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 Th12
      .= Partial_Sums((z+w) ExpSeq).k
    + ((Partial_Sums(( Expan_e(k+1,z,w) ))).k
    + (Alfa(k+1,z,w)).(k+1)) by A5;
 k+1-'(k+1)=0 by XREAL_1:232;
then  (Alfa(k+1,z,w)).(k+1)
    =(z ExpSeq).(k+1) * Partial_Sums(w ExpSeq).0 by Def11
      .=(z ExpSeq).(k+1) * ((w ExpSeq).0) by SERIES_1:def 1
      .=(z ExpSeq).(k+1) * 1 by Th10
      .=(Expan_e(k+1,z,w)).(k+1) by Th13;
then  (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 SERIES_1:def 1
      .=(z+w) |^ (k+1) /((k+1)! ) by Th8;
    then  Partial_Sums((Alfa(k+1,z,w))).(k+1) =Partial_Sums((z+w) ExpSeq).k
    +(z+w) ExpSeq.(k+1) by A6,Def4
      .=Partial_Sums((z+w) ExpSeq).(k+1) by SERIES_1:def 1;
    hence
Partial_Sums((z+w) ExpSeq).(k+1)=Partial_Sums(Alfa(k+1,z,w)).(k+1);
  end;
 for n holds X[n] from NAT_1:sch 2(A3,A4);
  hence thesis;
end;
