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

theorem
  for x being Real st x > 0 holds
    exp_R.x > x + 1 ::: shorter proof of strenghtened BOR_CANT:2
  proof
    let x be Real;
    assume
AA: x > 0;
    set r = 1;
    set f = Maclaurin(exp_R,].-r,r.[,x);
A4: exp_R.x = Sum f by TAYLOR_2:16;
A2: f.0 = x |^ 0 / (0!) by TAYLOR_2:8
       .= 1 by NEWTON:4,12;
A3: f.1 = x |^ 1 / (1!) by TAYLOR_2:8
       .= x by NEWTON:13;
SS: f is absolutely_summable by TAYLOR_2:16; then
A6: Sum f = (Partial_Sums f).1 + Sum (f ^\ (1 + 1)) by SERIES_1:15
         .= ((Partial_Sums f).0 + f.(0 + 1)) + Sum (f ^\ (1 + 1))
             by SERIES_1:def 1
         .= (1 + x) + Sum (f ^\ (1 + 1)) by A2,A3,SERIES_1:def 1;
    f is positive-yielding & f is summable by MacPositive,AA,SS;
    hence thesis by A4,A6,XREAL_1:29,Th36;
  end;
