Variables (K : {subfield L}) (x y : L).
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.

Lemma finite_PET : K_is_large \/ exists z, (<< <<K; y>>; x>> = <<K; z>>)%VS.
End FiniteCase.