In the axioms of a convexTvs, locally_convex should assume the convexity of a basis of 0-neighbourhoods, not of a basis of open sets.
locally_convex : exists2 B : set_system E,
(forall b, b \in B -> convex_set b) & basis B
should be
locally_convex : exists2 B : set_system E,
(forall b, b \in B -> convex_set b) & filter_from [set U | B U /\ U 0] id --> 0.
In the axioms of a convexTvs,
locally_convexshould assume the convexity of a basis of 0-neighbourhoods, not of a basis of open sets.locally_convex : exists2 B : set_system E, (forall b, b \in B -> convex_set b) & basis Bshould be
locally_convex : exists2 B : set_system E, (forall b, b \in B -> convex_set b) & filter_from [set U | B U /\ U 0] id --> 0.