Definition ss_ok (ss : SS.t) (interp: positive -> COINCpoint) :=
  forall s, SS.mem s ss = true ->
  forall cp, memCP cp s = true ->
    app coinc (interp_CP cp interp).

Lemma ss_ok_inter_ok1 : forall ss interp s1 s2 x (p : cartesianPower positive 1), ss_ok ss interp -> SS.In s1 ss -> pick_variety_aux (S.inter s1 s2) x = true -> S.mem p s1 = true -> app_1_n coinc (interp_CP p interp) (interp_CP x interp).