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.

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')).