theorem Th13:
  (z ExpSeq).k=(Expan_e(k,z,w)).k
proof
A1: 0 = k -k;
then A2: k-'k=0 by XREAL_1:233;
A3: (k-'k)! =1 by A1,Th1,XREAL_1:233;
  thus (Expan_e(k,z,w)).(k)=((Coef_e(k)).k) * (z |^ k) * (w |^ 0) by A2,Def10
    .=( (Coef_e(k)).k) * (z |^ k) * 1r by COMSEQ_3:11
    .=(1r/((k! ) * 1r)) * (z |^ k) by A3,Def7
    .=((z |^ k) * 1r)/(k! )
    .=(z ExpSeq).k by Def4;
end;
