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 p1 (pa x y) ∈ complex.

rewrite the current goal using quaternion_p1_eq x Hx y Hy (from left to right).

We will prove y ∈ complex.

An exact proof term for the current goal is Hy.

∎