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.

Lemma f_sup_X_eq_sup_image_f_X {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)
(CONTINUIOUS : isContinuous f)
(DIRECTED : isDirected X)
(SUPREMUM : is_supremum_of sup_X X)
(DIRECTED' : isDirected (E.image f X))
: f sup_X == supremum_cpo (E.image f X) DIRECTED'.