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 Th17:
  Expan_e(n,z,w) = (1r/(n!)) * Expan(n,z,w)
proof
  now
    let k be Element of NAT;
A1: now
      reconsider s = n! as Element of COMPLEX by XCMPLX_0:def 2;
A2:   1r/((k!) * ((n-'k)!)) =(((n!) * 1r)/(n!)) /((k!) * ((n-'k)!))
      by COMPLEX1:def 4,XCMPLX_1:60
        .= (1r/(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 Def3
        .= 1r/((k!) * ((n-'k)!)) * (z #N k) * (w #N (n-'k))
        by A3,SIN_COS:def 7;
      hence Expan_e(n,z,w).k = (1r/(n!)) * (n!) /((k!) * ((n-'k)!)) * ((z
      #N k) * (w #N (n-'k))) by A2,CLOPBAN3:38
        .= (1r/(n!)) * ((n!) /((k!) * ((n-'k)!))) * ((z #N k) * (w #N (n
      -'k))) by XCMPLX_1:74
        .= (1r/s) * ((s /((k!) * ((n-'k)!))) * ((z #N k) * (w #N (n-'k))))
        by CLOPBAN3:38
        .= (1r/s) * (s /((k!) * ((n-'k)!)) * (z #N k) * (w #N (n-' k)))
        by CLOPBAN3:38
        .= (1r/(n!)) * ((Coef(n)).k * ((z #N k)) * (w #N (n-'k)))
        by A3,SIN_COS:def 6
        .= (1r/(n!)) * Expan(n,z,w).k by A3,Def2
        .= ( (1r/(n!)) * Expan(n,z,w) ).k by CLVECT_1:def 14;
    end;
    now
      assume
A4:   n <k;
      hence Expan_e(n,z,w).k = 0.X by Def3
        .=(1r/(n!)) * 0.X by CLVECT_1:1
        .=(1r/(n!)) * Expan(n,z,w).k by A4,Def2
        .=((1r/(n!)) * Expan(n,z,w)).k by CLVECT_1:def 14;
    end;
    hence Expan_e(n,z,w).k = ( (1r/(n!)) * Expan(n,z,w) ).k by A1;
  end;
  hence thesis by FUNCT_2:63;
end;
