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.

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Lemma sin_plus : forall x y:R, sin (x + y) = sin x * cos y + cos x * sin y.