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 Th18:
  Expan_e(n,z,w)=(1/(n!)) * Expan(n,z,w)
proof
  now
    let k be Element of NAT;
A1: now
A2:   1/((k! ) * ((n-'k)! )) =(((n! ) * 1 )/(n! )) /((k! ) * ((n-'k)!)) by
XCMPLX_1:60
        .= (1 /(n! )) * (n! ) /((k! ) * ((n-'k)!)) by XCMPLX_1:74;
      assume
A3:   k <= n;
      hence Expan_e(n,z,w).k = (Coef_e(n)).k * (z #N k) * (w #N (n -' k))
      by Def7
        .= 1/( (k!) * ( (n -' k)!) ) * (z #N k) * (w #N (n -' k)) by A3,Def4;
      hence Expan_e(n,z,w).k = (1/(n! )) * (n! ) /((k! ) * ((n -' k)! )) * ((z
      #N k) * (w #N (n -' k))) by A2,LOPBAN_3:38
        .= (1/(n! )) * ( (n! ) /((k! ) * ((n -' k)! )) ) * ((z #N k) * (w #N
      (n -' k))) by XCMPLX_1:74
        .= (1/(n! )) * (( (n! ) /((k! ) * ((n -' k)! )) ) * ((z #N k) * (w
      #N (n -' k)))) by LOPBAN_3:38
        .= (1/(n! )) * ((n! ) /((k! ) * ((n -' k)! )) * (z #N k) * (w #N (n
      -' k))) by LOPBAN_3:38
        .= (1/(n! )) * ((Coef(n)).k * ((z #N k)) * (w #N (n -' k))) by A3,Def3
        .= (1/(n!)) * Expan(n,z,w).k by A3,Def6
        .= ( (1/(n!)) * Expan(n,z,w) ).k by NORMSP_1:def 5;
    end;
    now
      assume
A4:   n <k;
      hence Expan_e(n,z,w).k=0.X by Def7
        .=(1/(n!)) * 0.X by LOPBAN_3:38
        .=(1/(n!)) * Expan(n,z,w).k by A4,Def6
        .=((1/(n!)) * Expan(n,z,w)).k by NORMSP_1:def 5;
    end;
    hence Expan_e(n,z,w).k = ( (1/(n! )) * Expan(n,z,w) ).k by A1;
  end;
  hence thesis by FUNCT_2:63;
end;
