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 Th23:
  (z ExpSeq).k=(Expan_e(k,z,w)).k
proof
  0 = k -k;
  then
A1: k-'k=0 by XREAL_1:233;
  hence (Expan_e(k,z,w)).(k)=((Coef_e(k)).k) * (z #N k) * (w #N 0) by Def3
    .=( (Coef_e(k)).k) * (z #N k) * 1.X by CLOPBAN3:39
    .=( (Coef_e(k)).k) * (z #N k) by VECTSP_1:def 4
    .=(1r/((k!) * 1r)) * (z #N k) by A1,COMPLEX1:def 4,SIN_COS:1,def 7
    .=(z ExpSeq).k by Def1,COMPLEX1:def 4;
end;
