Section WeakTopology_and_Nets.
Variable I:DirectedSet.
Hypothesis I_nonempty: inhabited (DS_set I).
Variable x:Net I WeakTopology.
Variable x0:X.

(* Specifications: *)
Lemma net_limit_in_weak_topology_impl_net_limit_in_projections :
net_limit x x0 ->
forall a:A, net_limit (fun i:DS_set I => (f a) (x i)) ((f a) x0).
Lemma net_limit_in_projections_impl_net_limit_in_weak_topology :
(forall a:A, net_limit (fun i:DS_set I => (f a) (x i))
((f a) x0)) ->
net_limit x x0.