Let x0 of type ι → ο be given.
Assume H0:
∀ x1 . SNo x1 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x1) ⟶ x0 x2) ⟶ x0 x1.
Apply ordinal_ind with
λ x1 . ∀ x2 . x2 ∈ SNoS_ x1 ⟶ x0 x2.
Let x1 of type ι be given.
Assume H2:
∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ SNoS_ x2 ⟶ x0 x3.
Let x2 of type ι be given.
Assume H3:
x2 ∈ SNoS_ x1.
Apply SNoS_E2 with
x1,
x2,
x0 x2 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply H0 with
x2 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H2 with
SNoLev x2.
The subproof is completed by applying H4.
Apply SNo_ordinal_ind with
x0.
The subproof is completed by applying L1.