Let x0 of type ι → ι → CT2 ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι be given.
Assume H0:
∀ x5 . SNo x5 ⟶ ∀ x6 . SNo x6 ⟶ ∀ x7 x8 : ι → ι → ι . (∀ x9 . x9 ∈ SNoS_ (SNoLev x5) ⟶ ∀ x10 . SNo x10 ⟶ x7 x9 x10 = x8 x9 x10) ⟶ (∀ x9 . x9 ∈ SNoS_ (SNoLev x6) ⟶ x7 x5 x9 = x8 x5 x9) ⟶ x0 x5 x6 x7 = x0 x5 x6 x8.
Assume H1:
∀ x5 . x5 ∈ SNoS_ (SNoLev x1) ⟶ x2 x5 = x3 x5.
Assume H3:
∀ x5 . ordinal x5 ⟶ ∀ x6 . x6 ∈ SNoS_ x5 ⟶ SNo_rec_i (λ x7 . λ x8 : ι → ι . x0 x1 x7 (λ x9 x10 . If_i (x9 = x1) (x8 x10) (x2 x9 x10))) x6 = SNo_rec_i (λ x7 . λ x8 : ι → ι . x0 x1 x7 (λ x9 x10 . If_i (x9 = x1) (x8 x10) (x3 x9 x10))) x6.
Apply ordinal_ordsucc with
SNoLev x4.
Apply SNoLev_ordinal with
x4.
The subproof is completed by applying H2.
Apply SNoS_I with
ordsucc (SNoLev x4),
x4,
SNoLev x4 leaving 3 subgoals.
The subproof is completed by applying L4.
The subproof is completed by applying ordsuccI2 with
SNoLev x4.
Apply SNoLev_ with
x4.
The subproof is completed by applying H2.
Apply H3 with
ordsucc (SNoLev x4),
x4 leaving 2 subgoals.
The subproof is completed by applying L4.
The subproof is completed by applying L5.