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 Th7:
  Expan_e(n,z,w)=(1r/(n! )) (#) Expan(n,z,w)
proof
 now
    let k be Element of NAT;
A1: now
      assume
A2:   n <k;
      hence Expan_e(n,z,w).k =(1r/(n! )) * 0c by Def10
        .=(1r/(n! )) * Expan(n,z,w).k by A2,Def9
        .=((1r/(n! )) (#) Expan(n,z,w)).k by VALUED_1:6;
    end;
 now
      assume
A3:   k <= n;
then
A4:   Expan_e(n,z,w).k = (Coef_e(n)).k * (z |^ k) * (w |^ (n-'k)) by Def10
        .= 1r/((k! ) * ((n-'k)! )) * (z |^ k) * (w |^ (n-'k)) by A3,Def7;
   1r/((k! ) * ((n-'k)! )) =(((n! ) * 1r )/(n! )) /((k! ) * ((n-'k)!))
      by XCMPLX_1:60
        .= (1r /(n! )) * (n! ) /((k! ) * ((n-'k)! ));
      thus
      then Expan_e(n,z,w).k = (1r/(n! )) * (n! ) /((k! ) * ((n-'k)! ))
      * ((z |^ k) * (w |^ (n-'k))) by A4
        .= (1r/(n! )) * ((n! ) /((k! ) * ((n-'k)! ))
      * (z |^ k) * (w |^ (n-'k)))
        .= (1r/(n! )) * ((Coef(n)).k * ((z |^ k))
      * (w |^ (n-'k))) by A3,Def6
        .= (1r/(n! )) * Expan(n,z,w).k by A3,Def9
        .= ( (1r/(n! )) (#) Expan(n,z,w) ).k by VALUED_1:6;
    end;
    hence Expan_e(n,z,w).k = ( (1r/(n! )) (#) Expan(n,z,w) ).k by A1;
  end;
  hence thesis;
end;
