We prove the intermediate claim Li4: OSNo i4.
An exact proof term for the current goal is OSNo_Quaternion_k.
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 Li0: OSNo i0.
An exact proof term for the current goal is OSNo_Octonion_i0.
We prove the intermediate claim Li5i4: OSNo (i5 ⨯ i4).
An exact proof term for the current goal is OSNo_mul_OSNo i5 i4 ?? ??.
Apply OSNo_proj0proj1_split (i5 ⨯ i4) (:-: i0) ?? (OSNo_minus_OSNo i0 ??) to the current goal.
rewrite the current goal using minus_OSNo_proj0 i0 ?? (from left to right).
rewrite the current goal using OSNo_p0_i0 (from left to right).
rewrite the current goal using mul_OSNo_proj0 i5 i4 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_k (from left to right).
rewrite the current goal using OSNo_p1_k (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 * k + - (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 k HSNo_Quaternion_k (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 i0 ?? (from left to right).
rewrite the current goal using OSNo_p1_i0 (from left to right).
rewrite the current goal using mul_OSNo_proj1 i5 i4 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_k (from left to right).
rewrite the current goal using OSNo_p1_k (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) * k ' = - 1.
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using conj_HSNo_k (from left to right).
We will prove 0 + (- k) * (- k) = - 1.
rewrite the current goal using minus_mul_HSNo_distrL k (- k) HSNo_Quaternion_k (HSNo_minus_HSNo k HSNo_Quaternion_k) (from left to right).
rewrite the current goal using minus_mul_HSNo_distrR k k HSNo_Quaternion_k HSNo_Quaternion_k (from left to right).
We will prove 0 + - - k * k = - 1.
rewrite the current goal using Quaternion_k_sqr (from left to right).
rewrite the current goal using minus_HSNo_invol 1 HSNo_1 (from left to right).
An exact proof term for the current goal is add_HSNo_0L (- 1) (HSNo_minus_HSNo 1 HSNo_1).
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