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 Li4i5: OSNo (i4 ⨯ i5).
An exact proof term for the current goal is OSNo_mul_OSNo i4 i5 ?? ??.
Apply OSNo_proj0proj1_split (i4 ⨯ i5) i0 ?? ?? to the current goal.
rewrite the current goal using OSNo_p0_i0 (from left to right).
rewrite the current goal using mul_OSNo_proj0 i4 i5 ?? ?? (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 k * 0 + - ((- k) ' * 0) = 0.
rewrite the current goal using mul_HSNo_0R k HSNo_Quaternion_k (from left to right).
rewrite the current goal using mul_HSNo_0R ((- k) ') (HSNo_conj_HSNo (- 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 OSNo_p1_i0 (from left to right).
rewrite the current goal using mul_OSNo_proj1 i4 i5 ?? ?? (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 (- k) * k + 0 * 0 ' = 1.
rewrite the current goal using conj_HSNo_id_SNo 0 SNo_0 (from left to right).
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using minus_mul_HSNo_distrL k k HSNo_Quaternion_k HSNo_Quaternion_k (from left to right).
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_0R 1 HSNo_1.
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