We prove the intermediate claim Li2: OSNo i2.
An exact proof term for the current goal is OSNo_Quaternion_j.
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 Li0: OSNo i0.
An exact proof term for the current goal is OSNo_Octonion_i0.
We prove the intermediate claim Li6i2: OSNo (i6 ⨯ i2).
An exact proof term for the current goal is OSNo_mul_OSNo i6 i2 ?? ??.
Apply OSNo_proj0proj1_split (i6 ⨯ i2) (:-: 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 i6 i2 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_j (from left to right).
rewrite the current goal using OSNo_p1_j (from left to right).
rewrite the current goal using OSNo_p0_i6 (from left to right).
rewrite the current goal using OSNo_p1_i6 (from left to right).
We will prove 0 * j + - (0 ' * (- j)) = - 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 j HSNo_Quaternion_j (from left to right).
rewrite the current goal using mul_HSNo_0L (- j) (HSNo_minus_HSNo j HSNo_Quaternion_j) (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 i6 i2 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_j (from left to right).
rewrite the current goal using OSNo_p1_j (from left to right).
rewrite the current goal using OSNo_p0_i6 (from left to right).
rewrite the current goal using OSNo_p1_i6 (from left to right).
We will prove 0 * 0 + (- j) * j ' = - 1.
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using conj_HSNo_j (from left to right).
rewrite the current goal using minus_mul_HSNo_distrL j (- j) HSNo_Quaternion_j (HSNo_minus_HSNo j HSNo_Quaternion_j) (from left to right).
rewrite the current goal using minus_mul_HSNo_distrR j j HSNo_Quaternion_j HSNo_Quaternion_j (from left to right).
rewrite the current goal using Quaternion_j_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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