Section FiniteCase.
Variable N : nat.
Let K_is_large := exists s, [/\ uniq s, {subset s <= K} & N < size s].
Let cyclic_or_large (z : L) : z != 0 -> K_is_large \/ exists a, z ^+ a.+1 = 1.
End FiniteCase.
Hypothesis sepKy : separable_element K y.
End PrimitiveElementTheorem.

Lemma strong_Primitive_Element_Theorem K x y :
separable_element <<K; x>> y ->
exists2 z : L, (<< <<K; y>>; x>> = <<K; z>>)%VS
& separable_element K x -> separable_element K y.