assert (int' : 0 < PI2 < 2). {
destruct pi2_int; split; lra.
}
assert (lo2 := sin_pos_tech PI2 int').
assert (t2 : Rabs (sin PI2) = 1). {
rewrite <- Rabs_R1; apply Rsqr_eq_abs_0.
rewrite Rsqr_1, sin2, cos_pi2, Rsqr_0, Rminus_0_r; reflexivity.
}
revert t2; rewrite Rabs_pos_eq;[| apply Rlt_le]; tauto.
Qed.

Lemma sin_bound : forall (a : R) (n : nat), 0 <= a -> a <= PI ->
sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).