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 Li2: OSNo i2.
An exact proof term for the current goal is OSNo_Quaternion_j.
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 Li1i2: OSNo (i1 ⨯ i2).
An exact proof term for the current goal is OSNo_mul_OSNo i1 i2 ?? ??.
Apply OSNo_proj0proj1_split (i1 ⨯ i2) i4 ?? ?? to the current goal.
rewrite the current goal using OSNo_p0_k (from left to right).
rewrite the current goal using mul_OSNo_proj0 i1 i2 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_i (from left to right).
rewrite the current goal using OSNo_p1_i (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).
We will prove i * j + - (0 ' * 0) = k.
rewrite the current goal using conj_HSNo_id_SNo 0 SNo_0 (from left to right).
rewrite the current goal using mul_HSNo_0R 0 HSNo_0 (from left to right).
rewrite the current goal using minus_HSNo_0 (from left to right).
rewrite the current goal using add_HSNo_0R (i * j) (HSNo_mul_HSNo i j HSNo_Complex_i HSNo_Quaternion_j) (from left to right).
An exact proof term for the current goal is Quaternion_i_j.
rewrite the current goal using OSNo_p1_k (from left to right).
rewrite the current goal using mul_OSNo_proj1 i1 i2 ?? ?? (from left to right).
rewrite the current goal using OSNo_p0_i (from left to right).
rewrite the current goal using OSNo_p1_i (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).
We will prove 0 * i + 0 * j ' = 0.
rewrite the current goal using mul_HSNo_0L (j ') (HSNo_conj_HSNo j HSNo_Quaternion_j) (from left to right).
rewrite the current goal using mul_HSNo_0L i HSNo_Complex_i (from left to right).
An exact proof term for the current goal is add_HSNo_0L 0 HSNo_0.
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