Definition sub_silhouette {A} `{Perm_alg A} (a b: A) : Prop :=
    forall c, joins c b -> joins c a.Definition same_silhouette {A} `{Perm_alg A} (a b: A) : Prop :=
    forall c, joins c a <-> joins c b.

Lemma same_silhouette_join {A} `{HA: Perm_alg A}: forall phi phi' phiy phiz phiz', same_silhouette phi phi' -> join phi phiy phiz -> join phi' phiy phiz' -> same_silhouette phiz phiz'.