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 Li3: OSNo i3.
An exact proof term for the current goal is OSNo_Octonion_i3.
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 Li3i1: OSNo (i3 ⨯ i1).
An exact proof term for the current goal is OSNo_mul_OSNo i3 i1 ?? ??.
Apply OSNo_proj0proj1_split (i3 ⨯ i1) (:-: 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 i3 i1 ?? ?? (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_i3 (from left to right).
rewrite the current goal using OSNo_p1_i3 (from left to right).
We will prove 0 * i + - (0 ' * (- i)) = - 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 i HSNo_Complex_i (from left to right).
rewrite the current goal using mul_HSNo_0L (- i) (HSNo_minus_HSNo i HSNo_Complex_i) (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 i3 i1 ?? ?? (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_i3 (from left to right).
rewrite the current goal using OSNo_p1_i3 (from left to right).
We will prove 0 * 0 + (- i) * i ' = - 1.
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using conj_HSNo_i (from left to right).
rewrite the current goal using minus_mul_HSNo_distrL i (- i) HSNo_Complex_i (HSNo_minus_HSNo i HSNo_Complex_i) (from left to right).
rewrite the current goal using minus_mul_HSNo_distrR i i HSNo_Complex_i HSNo_Complex_i (from left to right).
rewrite the current goal using Quaternion_i_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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