Let z be given.

Assume Hz.

Apply quaternion_E z Hz to the current goal.

Let x be given.

Assume Hx.

Let y be given.

Assume Hy Hzxy.

rewrite the current goal using Hzxy (from left to right).

We will prove p0 (pa x y) ∈ complex.

rewrite the current goal using quaternion_p0_eq x Hx y Hy (from left to right).

We will prove x ∈ complex.

An exact proof term for the current goal is Hx.

∎