proposition: Lemma Qclose_separating_not_lt : forall q r : Q, (forall e, close e q r) -> ~ (q < r).
proof: Proof. intros q r E1 E2. pose proof (E1 (Qpos_diff _ _ E2)) as E3. apply symmetry in E3;apply Qclose_alt in E3. unfold cast in E3;simpl in E3. apply (irreflexivity lt (r - q)). apply le_lt_trans with (abs (r - q));trivial. apply Qabs_le_raw. Qed.
imports: HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.rationals HoTT.Classes.interfaces.orders HoTT.Classes.implementations.peano_naturals HoTT.Classes.implementations.natpair_integers HoTT.Classes.theory.groups HoTT.Classes.theory.integers HoTT.Classes.theory.dec_fields HoTT.Classes.orders.dec_fields HoTT.Classes.orders.sum HoTT.Classes.theory.rationals HoTT.Classes.orders.lattices HoTT.Classes.implementations.assume_rationals HoTT.Classes.tactics.ring_quote HoTT.Classes.tactics.ring_tac