We prove the intermediate claim Li1: OSNo i1.
An exact proof term for the current goal is OSNo_Complex_i.
We prove the intermediate claim Li5: OSNo i5.
An exact proof term for the current goal is OSNo_Octonion_i5.
We prove the intermediate claim Li6: OSNo i6.
An exact proof term for the current goal is OSNo_Octonion_i6.
We prove the intermediate claim Li5i1: OSNo (i5 ⨯ i1).
An exact proof term for the current goal is OSNo_mul_OSNo i5 i1 ?? ??.
Apply OSNo_proj0proj1_split (i5 ⨯ i1) (:-: i6) ?? (OSNo_minus_OSNo i6 ??) to the current goal.
rewrite the current goal using minus_OSNo_proj0 i6 ?? (from left to right).
rewrite the current goal using OSNo_p0_i6 (from left to right).
rewrite the current goal using mul_OSNo_proj0 i5 i1 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_i (from left to right).
rewrite the current goal using OSNo_p1_i (from left to right).
rewrite the current goal using OSNo_p0_i5 (from left to right).
rewrite the current goal using OSNo_p1_i5 (from left to right).
We will prove 0 * i + - (0 ' * (- k)) = - 0.
rewrite the current goal using conj_HSNo_id_SNo 0 SNo_0 (from left to right).
rewrite the current goal using mul_HSNo_0L i HSNo_Complex_i (from left to right).
rewrite the current goal using mul_HSNo_0L (- k) (HSNo_minus_HSNo k HSNo_Quaternion_k) (from left to right).
rewrite the current goal using minus_HSNo_0 (from left to right).
An exact proof term for the current goal is add_HSNo_0L 0 HSNo_0.
rewrite the current goal using minus_OSNo_proj1 i6 ?? (from left to right).
rewrite the current goal using OSNo_p1_i6 (from left to right).
rewrite the current goal using mul_OSNo_proj1 i5 i1 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_i (from left to right).
rewrite the current goal using OSNo_p1_i (from left to right).
rewrite the current goal using OSNo_p0_i5 (from left to right).
rewrite the current goal using OSNo_p1_i5 (from left to right).
We will prove 0 * 0 + (- k) * i ' = - (- j).
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using conj_HSNo_i (from left to right).
rewrite the current goal using minus_mul_HSNo_distrL k (- i) HSNo_Quaternion_k (HSNo_minus_HSNo i HSNo_Complex_i) (from left to right).
rewrite the current goal using minus_mul_HSNo_distrR k i HSNo_Quaternion_k HSNo_Complex_i (from left to right).
rewrite the current goal using Quaternion_k_i (from left to right).
rewrite the current goal using minus_HSNo_invol j HSNo_Quaternion_j (from left to right).
An exact proof term for the current goal is add_HSNo_0L j HSNo_Quaternion_j.
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