reserve X for 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 Th24:
  (z rExpSeq).k=(Expan_e(k,z,w)).k
proof
A1: 0 = k -k;
  then
A2: (k-'k)! =1 by NEWTON:12,XREAL_1:233;
  k-'k=0 by A1,XREAL_1:233;
  hence (Expan_e(k,z,w)).(k)=((Coef_e(k)).k) * (z #N k) * (w #N 0) by Def7
    .=( (Coef_e(k)).k) * (z #N k) * 1.X by LOPBAN_3:39
    .=( (Coef_e(k)).k) * (z #N k) by LOPBAN_3:38
    .=(1/((k!) * 1)) * (z #N k) by A2,Def4
    .=(z rExpSeq).k by Def2;
end;
