Section CPO_THEORY.
#[local] Existing Instance direct_product_of_two_Prosets.
#[local] Existing Instance subProset.
#[local] Existing Instance pi_isProset.
#[local] Existing Instance Scott_topology.
Definition U {D : Type} {PROSET : isProset D} (x : D) : ensemble D :=
fun z : D => ~ z =< x.
#[local] Hint Resolve ScottContinuousMap_isMonotonic : poset_hints.
Proof with eauto with *.
assert (MONOTONIC : isMonotonic1 f) by now eapply ScottContinuousMap_isMonotonic.
set (E.image f X) as Y in *. set (sup_Y := supremum_cpo Y DIRECTED').
pose proof (SUPREMUM' := supremum_cpo_spec Y DIRECTED').
assert (claim1 : sup_Y =< f sup_X).
{ eapply SUPREMUM'. intros y y_in_Y. unfold Y in *. s!.
des. subst y. eapply MONOTONIC, SUPREMUM...
}
assert (claim2 : f sup_X =< sup_Y).
{ eapply NNPP. intros f_sup_X_in_U_sup_Y.
assert (sup_X_in_preimage_f_U_sup_Y : sup_X \in E.preimage f (U sup_Y)) by now done!.
assert (f_U_sup_Y_isOpen : isOpen (E.preimage f (U sup_Y))) by now eapply CONTINUIOUS, U_x_isOpen.
inv f_U_sup_Y_isOpen. pose proof (LIMIT X sup_X DIRECTED SUPREMUM sup_X_in_preimage_f_U_sup_Y) as [x1 [x1_in_X x1_in_preimage_f_U_sup_Y]].
assert (f_x1_in_image_f_X : f x1 \in E.image f X).
{ econstructor... }
assert (f_x1_in_U_sup_Y : f x1 \in U sup_Y).
{ s!. des. done!. }
contradiction f_x1_in_U_sup_Y. eapply SUPREMUM'...
}
eapply @leProp_antisymmetry with (A_isProset := PROSET')...
Qed.

Lemma sup_Y_is_supremum_of_image_f_X_iff_f_sup_X_eq_sup_Y {D : Type} {D' : Type} {PROSET : isProset D} {PROSET' : isProset D'} {CPO : isCpo D} {CPO' : isCpo D'} (f : D -> D') (X : ensemble D) (sup_X : D) (sup_Y : D')
(CONTINUIOUS : isContinuous f)
(DIRECTED : isDirected X)
(SUPREMUM : is_supremum_of sup_X X)
: is_supremum_of sup_Y (E.image f X) <-> f sup_X == sup_Y.