Corollary UAL-7. A ruler construction for cubic and conic. The type is ⟨5, 2, 0⟩.

\{ \begin{matrix}full symmetry of f \\ f(e, e, e, e, e) = e \\ f(e, e, e, x, f(e, e, e, x, y)) = y \\ h(x, y) = h(y, x) \\ h(e, e) = e \\ h(x, h(x, y)) = y \end{matrix} \} =(gL)⇒

{h(h(h(x, y), h(z, u)), v) = f(x, y, z, u, v)}.

Proof. This follows from Thm. UAL-5. Define

g(x, y, z, u, v) = h(h(h(x, y), h(z, u)), v).

Then g satisfies the hypotheses of Thm. UAL-6.