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Proofgold Proof

pf
Claim L0: OSNo Octonion_i0
The subproof is completed by applying OSNo_Octonion_i0.
Claim L1: OSNo Quaternion_k
The subproof is completed by applying OSNo_Quaternion_k.
Claim L2: OSNo Octonion_i5
The subproof is completed by applying OSNo_Octonion_i5.
Claim L3: OSNo (mul_OSNo Octonion_i0 Quaternion_k)
Apply OSNo_mul_OSNo with Octonion_i0, Quaternion_k leaving 2 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L1.
Apply OSNo_proj0proj1_split with mul_OSNo Octonion_i0 Quaternion_k, Octonion_i5 leaving 4 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L2.
Apply OSNo_p0_i5 with λ x0 x1 . OSNo_proj0 (mul_OSNo Octonion_i0 Quaternion_k) = x1.
Apply mul_OSNo_proj0 with Octonion_i0, Quaternion_k, λ x0 x1 . x1 = 0 leaving 3 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L1.
Apply OSNo_p0_i0 with λ x0 x1 . add_HSNo (mul_HSNo x1 (OSNo_proj0 Quaternion_k)) (minus_HSNo (mul_HSNo (conj_HSNo (OSNo_proj1 Quaternion_k)) (OSNo_proj1 Octonion_i0))) = 0.
Apply OSNo_p1_i0 with λ x0 x1 . add_HSNo (mul_HSNo 0 (OSNo_proj0 Quaternion_k)) (minus_HSNo (mul_HSNo (conj_HSNo (OSNo_proj1 Quaternion_k)) x1)) = 0.
Apply OSNo_p0_k with λ x0 x1 . add_HSNo (mul_HSNo 0 x1) (minus_HSNo (mul_HSNo (conj_HSNo (OSNo_proj1 Quaternion_k)) 1)) = 0.
Apply OSNo_p1_k with λ x0 x1 . add_HSNo (mul_HSNo 0 Quaternion_k) (minus_HSNo (mul_HSNo (conj_HSNo x1) 1)) = 0.
Apply conj_HSNo_id_SNo with 0, λ x0 x1 . add_HSNo (mul_HSNo 0 Quaternion_k) (minus_HSNo (mul_HSNo x1 1)) = 0 leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply mul_HSNo_0L with Quaternion_k, λ x0 x1 . add_HSNo x1 (minus_HSNo (mul_HSNo 0 1)) = 0 leaving 2 subgoals.
The subproof is completed by applying HSNo_Quaternion_k.
Apply mul_HSNo_0L with 1, λ x0 x1 . add_HSNo 0 (minus_HSNo x1) = 0 leaving 2 subgoals.
The subproof is completed by applying HSNo_1.
Apply minus_HSNo_0 with λ x0 x1 . add_HSNo 0 x1 = 0.
Apply add_HSNo_0L with 0.
The subproof is completed by applying HSNo_0.
Apply OSNo_p1_i5 with λ x0 x1 . OSNo_proj1 (mul_OSNo Octonion_i0 Quaternion_k) = x1.
Apply mul_OSNo_proj1 with Octonion_i0, Quaternion_k, λ x0 x1 . x1 = minus_HSNo Quaternion_k leaving 3 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L1.
Apply OSNo_p0_i0 with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj1 Quaternion_k) x1) (mul_HSNo (OSNo_proj1 Octonion_i0) (conj_HSNo (OSNo_proj0 Quaternion_k))) = minus_HSNo Quaternion_k.
Apply OSNo_p1_i0 with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj1 Quaternion_k) 0) (mul_HSNo x1 (conj_HSNo (OSNo_proj0 Quaternion_k))) = minus_HSNo Quaternion_k.
Apply OSNo_p0_k with λ x0 x1 . add_HSNo (mul_HSNo (OSNo_proj1 Quaternion_k) 0) (mul_HSNo 1 (conj_HSNo x1)) = minus_HSNo Quaternion_k.
Apply OSNo_p1_k with λ x0 x1 . add_HSNo (mul_HSNo x1 0) (mul_HSNo 1 (conj_HSNo Quaternion_k)) = minus_HSNo Quaternion_k.
Apply mul_HSNo_0L with 0, λ x0 x1 . add_HSNo x1 (mul_HSNo 1 (conj_HSNo Quaternion_k)) = minus_HSNo Quaternion_k leaving 2 subgoals.
The subproof is completed by applying HSNo_0.
Apply mul_HSNo_1L with conj_HSNo Quaternion_k, λ x0 x1 . add_HSNo 0 x1 = minus_HSNo Quaternion_k leaving 2 subgoals.
Apply HSNo_conj_HSNo with Quaternion_k.
The subproof is completed by applying HSNo_Quaternion_k.
Apply conj_HSNo_k with λ x0 x1 . add_HSNo 0 x1 = minus_HSNo Quaternion_k.
Apply add_HSNo_0L with minus_HSNo Quaternion_k.
Apply HSNo_minus_HSNo with Quaternion_k.
The subproof is completed by applying HSNo_Quaternion_k.
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