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 Li1i3: OSNo (i1 ⨯ i3).
An exact proof term for the current goal is OSNo_mul_OSNo i1 i3 ?? ??.
Apply OSNo_proj0proj1_split (i1 ⨯ i3) i0 ?? ?? to the current goal.
rewrite the current goal using OSNo_p0_i0 (from left to right).
rewrite the current goal using mul_OSNo_proj0 i1 i3 ?? ?? (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 i * 0 + - ((- i) ' * 0) = 0.
rewrite the current goal using mul_HSNo_0R i HSNo_Complex_i (from left to right).
rewrite the current goal using mul_HSNo_0R ((- i) ') (HSNo_conj_HSNo (- 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 OSNo_p1_i0 (from left to right).
rewrite the current goal using mul_OSNo_proj1 i1 i3 ?? ?? (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 (- i) * i + 0 * 0 ' = 1.
rewrite the current goal using conj_HSNo_id_SNo 0 SNo_0 (from left to right).
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using minus_mul_HSNo_distrL i i HSNo_Complex_i HSNo_Complex_i (from left to right).
We will prove - i * i + 0 = 1.
rewrite the current goal using Quaternion_i_sqr (from left to right).
We will prove - - 1 + 0 = 1.
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_0R 1 HSNo_1.
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