proposition: Lemma inf_is_low_bd : ∀ A : (ℝ → Prop), ∀ m : ℝ, is_inf A m ⇒ is_lower_bound A m.
proof: Proof. Take A : (ℝ → Prop). Take m : R. Assume that (is_inf A m) (i). Because (i) both (is_lower_bound A m) (ii) and (for all M0 : ℝ, is_lower_bound A M0 ⇨ M0 ≤ m) hold. By (ii) we conclude that (is_lower_bound A m). Qed.
imports: Classical Classical_Pred_Type Coq.Reals.Reals Automation Notations Tactics