 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem MacPositive:
  for x,r being Real st x > 0 & r > 0 holds
    Maclaurin(exp_R,].-r,r.[,x) is positive-yielding
  proof
    let x,r be Real;
    assume
A0: x > 0 & r > 0;
    set f = Maclaurin(exp_R,].-r,r.[,x);
    for r being Real st r in rng f holds 0 < r
    proof
      let r be Real;
      assume r in rng f; then
      consider xx being object such that
A1:   xx in dom f & r = f.xx by FUNCT_1:def 3;
      reconsider nn = xx as Element of NAT by A1;
      r = x |^ nn / (nn!) by A1,TAYLOR_2:8,A0;
      hence thesis by A0;
    end;
    hence thesis by PARTFUN3:def 1;
  end;
