rewrite the current goal using minus_OSNo_minus_HSNo 1 HSNo_1 (from left to right).
We prove the intermediate claim Li0i0: OSNo (i0 ⨯ i0).
An exact proof term for the current goal is OSNo_mul_OSNo i0 i0 OSNo_Octonion_i0 OSNo_Octonion_i0.
We will prove i0 ⨯ i0 = - 1.
We prove the intermediate claim L1: HSNo (- 1).
An exact proof term for the current goal is HSNo_minus_HSNo 1 HSNo_1.
Apply OSNo_proj0proj1_split (i0 ⨯ i0) (- 1) Li0i0 (HSNo_OSNo (- 1) L1) to the current goal.
We will prove p0 (i0 ⨯ i0) = p0 (- 1).
rewrite the current goal using HSNo_OSNo_proj0 (- 1) L1 (from left to right).
We will prove p0 (i0 ⨯ i0) = - 1.
rewrite the current goal using mul_OSNo_proj0 i0 i0 OSNo_Octonion_i0 OSNo_Octonion_i0 (from left to right).
rewrite the current goal using OSNo_p0_i0 (from left to right).
rewrite the current goal using OSNo_p1_i0 (from left to right).
We will prove 0 * 0 + - (1 ' * 1) = - 1.
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
We will prove 0 + - (1 ' * 1) = - 1.
rewrite the current goal using conj_HSNo_id_SNo 1 SNo_1 (from left to right).
We will prove 0 + - (1 * 1) = - 1.
rewrite the current goal using mul_HSNo_1L 1 HSNo_1 (from left to right).
We will prove 0 + - 1 = - 1.
An exact proof term for the current goal is add_HSNo_0L (- 1) L1.
We will prove p1 (i0 ⨯ i0) = p1 (- 1).
rewrite the current goal using HSNo_OSNo_proj1 (- 1) L1 (from left to right).
We will prove p1 (i0 ⨯ i0) = 0.
rewrite the current goal using mul_OSNo_proj1 i0 i0 OSNo_Octonion_i0 OSNo_Octonion_i0 (from left to right).
We will prove p1 i0 * p0 i0 + p1 i0 * p0 i0 ' = 0.
rewrite the current goal using OSNo_p0_i0 (from left to right).
rewrite the current goal using OSNo_p1_i0 (from left to right).
We will prove 1 * 0 + 1 * 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 1 HSNo_1 (from left to right).
An exact proof term for the current goal is add_HSNo_0L 0 HSNo_0.
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