rewrite the current goal using minus_OSNo_minus_HSNo 1 HSNo_1 (from left to right).
We prove the intermediate claim Li6i6: OSNo (i6 ⨯ i6).
An exact proof term for the current goal is OSNo_mul_OSNo i6 i6 OSNo_Octonion_i6 OSNo_Octonion_i6.
We will prove i6 ⨯ i6 = - 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 (i6 ⨯ i6) (- 1) Li6i6 (HSNo_OSNo (- 1) L1) to the current goal.
We will prove p0 (i6 ⨯ i6) = p0 (- 1).
rewrite the current goal using HSNo_OSNo_proj0 (- 1) L1 (from left to right).
We will prove p0 (i6 ⨯ i6) = - 1.
rewrite the current goal using mul_OSNo_proj0 i6 i6 OSNo_Octonion_i6 OSNo_Octonion_i6 (from left to right).
rewrite the current goal using OSNo_p0_i6 (from left to right).
rewrite the current goal using OSNo_p1_i6 (from left to right).
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
We will prove 0 + - ((- j) ' * (- j)) = - 1.
rewrite the current goal using conj_minus_HSNo j HSNo_Quaternion_j (from left to right).
We will prove 0 + - ((- j ') * (- j)) = - 1.
rewrite the current goal using conj_HSNo_j (from left to right).
We will prove 0 + - ((- - j) * (- j)) = - 1.
rewrite the current goal using minus_HSNo_invol j HSNo_Quaternion_j (from left to right).
We will prove 0 + - (j * (- j)) = - 1.
rewrite the current goal using minus_mul_HSNo_distrR j j HSNo_Quaternion_j HSNo_Quaternion_j (from left to right).
We will prove 0 + - - (j * j) = - 1.
rewrite the current goal using Quaternion_j_sqr (from left to right).
rewrite the current goal using minus_HSNo_invol 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 (i6 ⨯ i6) = p1 (- 1).
rewrite the current goal using HSNo_OSNo_proj1 (- 1) L1 (from left to right).
We will prove p1 (i6 ⨯ i6) = 0.
rewrite the current goal using mul_OSNo_proj1 i6 i6 OSNo_Octonion_i6 OSNo_Octonion_i6 (from left to right).
We will prove p1 i6 * p0 i6 + p1 i6 * p0 i6 ' = 0.
rewrite the current goal using OSNo_p0_i6 (from left to right).
rewrite the current goal using OSNo_p1_i6 (from left to right).
We will prove (- j) * 0 + (- j) * 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 (- j) (HSNo_minus_HSNo j HSNo_Quaternion_j) (from left to right).
An exact proof term for the current goal is add_HSNo_0L 0 HSNo_0.
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