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.

Lemma U_x_isOpen {D : Type} {PROSET : isProset D} (x : D)
: isOpen (U x).