Let x0 of type ι be given.

Assume H0: x0 ∈ octonion.

Let x1 of type ο be given.

Assume H1: ∀ x2 . x2 ∈ quaternion ⟶ ∀ x3 . x3 ∈ quaternion ⟶ x0 = HSNo_pair x2 x3 ⟶ x1.

Apply ReplE_impred with setprod quaternion quaternion, λ x2 . HSNo_pair (ap x2 0) (ap x2 1), x0, x1 leaving 2 subgoals.

The subproof is completed by applying H0.

Let x2 of type ι be given.

Assume H3: x0 = HSNo_pair (ap x2 0) (ap x2 1).

Apply H1 with ap x2 0, ap x2 1 leaving 3 subgoals.

Apply ap0_Sigma with quaternion, λ x3 . quaternion, x2.

The subproof is completed by applying H2.

Apply ap1_Sigma with quaternion, λ x3 . quaternion, x2.

The subproof is completed by applying H2.

The subproof is completed by applying H3.

■

Proofgold Proof