Let x0 of type ι be given.
Apply mul_SNo_eq_2 with
x0,
0,
mul_SNo x0 0 = 0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply H7 with
λ x3 x4 . x4 = 0.
Claim L9: x2 = 0
Apply Empty_Subq_eq with
x2.
Let x3 of type ι be given.
Assume H9: x3 ∈ x2.
Apply FalseE with
x3 ∈ 0.
Apply H4 with
x3,
False leaving 3 subgoals.
The subproof is completed by applying H9.
Let x4 of type ι be given.
Assume H10:
x4 ∈ SNoL x0.
Let x5 of type ι be given.
Apply SNoR_0 with
λ x6 x7 . ... ⟶ ... = ... ⟶ False.
Apply L8 with
λ x3 x4 . SNoCut x4 x2 = 0.
Apply L9 with
λ x3 x4 . SNoCut 0 x4 = 0.
The subproof is completed by applying SNoCut_0_0.