We prove the intermediate claim Li6: OSNo i6.
An exact proof term for the current goal is OSNo_Octonion_i6.
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 Li4: OSNo i4.
An exact proof term for the current goal is OSNo_Quaternion_k.
We prove the intermediate claim Li3i6: OSNo (i3 ⨯ i6).
An exact proof term for the current goal is OSNo_mul_OSNo i3 i6 ?? ??.
Apply OSNo_proj0proj1_split (i3 ⨯ i6) (:-: i4) ?? (OSNo_minus_OSNo i4 ??) to the current goal.
rewrite the current goal using minus_OSNo_proj0 i4 ?? (from left to right).
rewrite the current goal using OSNo_p0_k (from left to right).
rewrite the current goal using mul_OSNo_proj0 i3 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 OSNo_p0_i3 (from left to right).
rewrite the current goal using OSNo_p1_i3 (from left to right).
We will prove 0 * 0 + - ((- j) ' * (- i)) = - k.
rewrite the current goal using mul_HSNo_0L 0 HSNo_0 (from left to right).
rewrite the current goal using conj_minus_HSNo j HSNo_Quaternion_j (from left to right).
rewrite the current goal using conj_HSNo_j (from left to right).
rewrite the current goal using minus_HSNo_invol j HSNo_Quaternion_j (from left to right).
We will prove 0 + - (j * (- i)) = - k.
rewrite the current goal using minus_mul_HSNo_distrR j i HSNo_Quaternion_j HSNo_Complex_i (from left to right).
rewrite the current goal using Quaternion_j_i (from left to right).
rewrite the current goal using minus_HSNo_invol k HSNo_Quaternion_k (from left to right).
An exact proof term for the current goal is add_HSNo_0L (- k) (HSNo_minus_HSNo k HSNo_Quaternion_k).
rewrite the current goal using minus_OSNo_proj1 i4 ?? (from left to right).
rewrite the current goal using OSNo_p1_k (from left to right).
rewrite the current goal using mul_OSNo_proj1 i3 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 OSNo_p0_i3 (from left to right).
rewrite the current goal using OSNo_p1_i3 (from left to right).
We will prove (- j) * 0 + (- i) * 0 ' = - 0.
rewrite the current goal using conj_HSNo_id_SNo 0 SNo_0 (from left to right).
rewrite the current goal using minus_HSNo_0 (from left to right).
rewrite the current goal using mul_HSNo_0R (- i) (HSNo_minus_HSNo i HSNo_Complex_i) (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.
∎