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 Li1: OSNo i1.
An exact proof term for the current goal is OSNo_Complex_i.
We prove the intermediate claim Li5i6: OSNo (i5 ⨯ i6).
An exact proof term for the current goal is OSNo_mul_OSNo i5 i6 ?? ??.
Apply OSNo_proj0proj1_split (i5 ⨯ i6) i1 ?? ?? to the current goal.
rewrite the current goal using OSNo_p0_i (from left to right).
rewrite the current goal using mul_OSNo_proj0 i5 i6 ?? ?? (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).
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) ' * (- k)) = i.
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using conj_minus_HSNo j HSNo_Quaternion_j (from left to right).
rewrite the current goal using conj_HSNo_j (from left to right).
rewrite the current goal using minus_HSNo_invol j HSNo_Quaternion_j (from left to right).
We will prove 0 + - (j * (- k)) = i.
rewrite the current goal using minus_mul_HSNo_distrR j k HSNo_Quaternion_j HSNo_Quaternion_k (from left to right).
rewrite the current goal using Quaternion_j_k (from left to right).
rewrite the current goal using minus_HSNo_invol i HSNo_Complex_i (from left to right).
An exact proof term for the current goal is add_HSNo_0L i HSNo_Complex_i.
rewrite the current goal using OSNo_p1_i (from left to right).
rewrite the current goal using mul_OSNo_proj1 i5 i6 ?? ?? (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).
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 (- j) * 0 + (- k) * 0 ' = 0.
rewrite the current goal using conj_HSNo_id_SNo 0 SNo_0 (from left to right).
rewrite the current goal using mul_HSNo_0R (- k) (HSNo_minus_HSNo k HSNo_Quaternion_k) (from left to right).
rewrite the current goal using mul_HSNo_0R (- j) (HSNo_minus_HSNo j HSNo_Quaternion_j) (from left to right).
An exact proof term for the current goal is add_HSNo_0L 0 HSNo_0.
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