reserve Z for open Subset of REAL;

theorem
  for n be Element of NAT, r,x be Real st 0 < r holds Maclaurin(exp_R,
  ].-r,r.[,x).n = x |^ n / (n!)
proof
  let n be Element of NAT;
A1: |.0-0.|=0 by ABSVALUE:2;
  let r,x be Real;
  assume r > 0;
  then 0 in ].0-r,0+r.[ by A1,RCOMP_1:1;
  then
A2: 0 in dom(exp_R | ].-r,r.[) by Th5;
  Maclaurin(exp_R, ].-r,r.[,x).n = (diff(exp_R,].-r,r.[).n).0 * (x-0) |^ n
  / (n!) by TAYLOR_1:def 7
    .=(exp_R | ].-r,r.[).0 * x |^ n / (n!) by Th6
    .=exp_R.0 * x |^ n / (n!) by A2,FUNCT_1:47
    .=x |^ n / (n!) by SIN_COS2:13;
  hence thesis;
end;
