Section Coinc_refl.
Context {AR : Arity}.
Definition pick_variety_auxCP {m : nat} (s : SS.elt) (cp : cartesianPower positive (S (S m))) : bool.
Definition pick_variety (s : SS.elt) (st : STt) :=
STexists_ (fun t => pick_variety_aux s t) st.

Lemma proper_1 : forall s1 st,
Proper (S.Equal ==> eq)
(fun s2 : SS.elt => pick_variety (S.inter s1 s2) st).