We prove the intermediate claim Li3: OSNo i3.
An exact proof term for the current goal is OSNo_Octonion_i3.
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 Li2: OSNo i2.
An exact proof term for the current goal is OSNo_Quaternion_j.
We prove the intermediate claim Li3i5: OSNo (i3 ⨯ i5).
An exact proof term for the current goal is OSNo_mul_OSNo i3 i5 ?? ??.
Apply OSNo_proj0proj1_split (i3 ⨯ i5) i2 ?? ?? to the current goal.
rewrite the current goal using OSNo_p0_j (from left to right).
rewrite the current goal using mul_OSNo_proj0 i3 i5 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_i3 (from left to right).
rewrite the current goal using OSNo_p1_i3 (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_minus_HSNo k HSNo_Quaternion_k (from left to right).
rewrite the current goal using conj_HSNo_k (from left to right).
rewrite the current goal using minus_HSNo_invol k HSNo_Quaternion_k (from left to right).
We will prove 0 + - (k * (- i)) = j.
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.
rewrite the current goal using OSNo_p1_j (from left to right).
rewrite the current goal using mul_OSNo_proj1 i3 i5 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_i3 (from left to right).
rewrite the current goal using OSNo_p1_i3 (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 + (- i) * 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 (- i) (HSNo_minus_HSNo i HSNo_Complex_i) (from left to right).
rewrite the current goal using mul_HSNo_0R (- k) (HSNo_minus_HSNo k HSNo_Quaternion_k) (from left to right).
An exact proof term for the current goal is add_HSNo_0L 0 HSNo_0.
∎