reserve Z for open Subset of REAL;

theorem
  for r, x be Real st 0 < r holds Maclaurin(exp_R,].-r,r.[,x) = (x
  rExpSeq) & Maclaurin(exp_R,].-r,r.[,x) is absolutely_summable & exp_R.x=Sum(
  Maclaurin(exp_R,].-r,r.[,x))
proof
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;
  now
    let t be object;
    assume t in NAT;
    then reconsider n=t as Element of NAT;
    thus Maclaurin(exp_R,].-r,r.[,x).t = (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 rExpSeq).t by SIN_COS:def 5,SIN_COS2:13;
  end;
  then Maclaurin(exp_R,].-r,r.[,x) = (x rExpSeq) by FUNCT_2:12;
  hence thesis by Th15,SIN_COS:def 22;
end;
