proposition: Lemma Pr_union E1 E2 : Pr (E1 :|: E2) <= Pr E1 + Pr E2.
proof: Proof. rewrite /Pr. rewrite [X in X <= _](_ : _ = \sum_(i in A | [predU E1 & E2] i) P i); last first. by apply eq_bigl => x /=; rewrite inE. rewrite [X in _ <= X + _](_ : _ = \sum_(i in A | pred_of_set E1 i) P i); last first. by apply eq_bigl => x /=; rewrite unfold_in. rewrite [X in _ <= _ + X](_ : _ = \sum_(i in A | pred_of_set E2 i) P i); last first. by apply eq_bigl => x /=; rewrite unfold_in. exact/leR_sumR_predU. Qed.
imports: mathcomp(all_ssreflect ssralg ssrnum matrix) mathcomp(Rstruct reals) Reals Lra ssrR Reals_ext realType_ext logb ssr_ext ssralg_ext bigop_ext fdist