Definition ps_atan_exists_1 (x : R) (Hx : -1 <= x <= 1) :
   {l : R | Un_cv (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}.Definition in_int (x : R) : {-1 <= x <= 1}+{~ -1 <= x <= 1}.Definition ps_atan (x : R) : R :=
 match in_int x with
   left h => let (v, _) := ps_atan_exists_1 x h in v
 | right h => atan x
 end.

(* Specifications: *)
Lemma ps_atan0_0 : ps_atan 0 = 0.
Lemma ps_atan_exists_1_opp : forall x h h', proj1_sig (ps_atan_exists_1 (-x) h) = -(proj1_sig (ps_atan_exists_1 x h')).
Lemma ps_atan_opp : forall x, ps_atan (-x) = -ps_atan x.
Lemma ps_atanSeq_continuity_pt_1 : forall (N : nat) (x : R), 0 <= x -> x <= 1 -> continuity_pt (fun x => sum_f_R0 (tg_alt (Ratan_seq x)) N) x.