Section Lemmas_on_partial_orders.
Variable U : Type.
Variable B : Ensemble U.
Variable D : PO U.
Hint Resolve Rel_of_reflexive.
Hint Resolve Rel_of_antisymmetric.
Hint Resolve Rel_of_transitive.
Hint Resolve Couple_included_in_carrier.
Hint Resolve Singleton_has_lub.
Hint Resolve Empty_set_has_Upper_Bound.
intros x H'; apply Upper_Bound_definition; auto with sets.
intros y H'0; elim H'0; auto with sets.
Qed.
Hint Resolve Anyone_is_Upper_Bound_of_Empty_set.
Hint Resolve Empty_set_has_lub.
Hint Resolve Conditionally_complete_has_a_bottom.
Hint Resolve Bottom_is_compact.
Hint Resolve Compact_is_in_Carrier.
Hint Resolve Compatible_is_reflexive.
Hint Immediate Compatible_is_symmetric.
intros x y H'; red in |- *; simpl in |- *.
intros H'0 x0 y0 H'1; try assumption.
red in H'1.
lapply (H'1 x0); [ intro H'3 | auto with sets ].
lapply (H'1 y0); [ intro H'4 | auto with sets ].
elim H'4.
elim H'3; auto with sets.
elim H'3; auto with sets.
Qed.
Hint Resolve Coherent_implies_Conditionally_Complete.

Theorem Coherent_has_a_bottom :
Coherent U D -> exists bot : U, Bottom U D bot.