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 Th13:
  (z ExpSeq).k=(Expan_e(k,z,w)).k
proof
A1: 0 = k -k;
then A2: k-'k=0 by XREAL_1:233;
A3: (k-'k)! =1 by A1,Th1,XREAL_1:233;
  thus (Expan_e(k,z,w)).(k)=((Coef_e(k)).k) * (z |^ k) * (w |^ 0) by A2,Def10
    .=( (Coef_e(k)).k) * (z |^ k) * 1r by COMSEQ_3:11
    .=(1r/((k! ) * 1r)) * (z |^ k) by A3,Def7
    .=((z |^ k) * 1r)/(k! )
    .=(z ExpSeq).k by Def4;
end;
