Let x0 of type ι → ι → CT2 ι be given.
Assume H0:
∀ x1 . SNo x1 ⟶ ∀ x2 . SNo x2 ⟶ ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x6 . SNo x6 ⟶ x3 x5 x6 = x4 x5 x6) ⟶ (∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x3 x1 x5 = x4 x1 x5) ⟶ x0 x1 x2 x3 = x0 x1 x2 x4.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Claim L2:
∀ x3 . SNo x3 ⟶ ∀ x4 x5 : ι → ι . (∀ x6 . x6 ∈ SNoS_ (SNoLev x3) ⟶ x4 x6 = x5 x6) ⟶ (λ x6 . λ x7 : ι → ι → ι . λ x8 . λ x9 : ι → ι . x0 x6 x8 (λ x10 x11 . If_i (x10 = x6) (x9 x11) (x7 x10 x11))) x1 x2 x3 x4 = (λ x6 . λ x7 : ι → ι → ι . λ x8 . λ x9 : ι → ι . x0 x6 x8 (λ x10 x11 . If_i (x10 = x6) (x9 x11) (x7 x10 x11))) x1 x2 x3 x5Let x3 of type ι be given.
Let x4 of type ι → ι be given.
Let x5 of type ι → ι be given.
Assume H3:
∀ x6 . x6 ∈ SNoS_ (SNoLev x3) ⟶ x4 x6 = x5 x6.
Claim L4:
∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ x2 x6 = x2 x6Let x6 of type ι be given.
Let x7 of type (ι → ι) → (ι → ι) → ο be given.
Assume H5: x7 (x2 x6) (x2 x6).
The subproof is completed by applying H5.
Apply SNo_rec2_G_prop with
x0,
x1,
x2,
x2,
x3,
x4,
x5 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying L4.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply SNo_rec_i_eq with
(λ x3 . λ x4 : ι → ι → ι . λ x5 . λ x6 : ι → ι . x0 x3 x5 (λ x7 x8 . If_i (x7 = x3) (x6 x8) (x4 x7 x8))) x1 x2.
The subproof is completed by applying L2.