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 be given.
Claim L4:
(λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (SNo x5) (SNo_rec_i ((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))) x3 x4) x5) 0) x1 (SNo_rec_ii (λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (SNo x5) (SNo_rec_i ((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))) x3 x4) x5) 0)) x2 = x0 x1 x2 (SNo_rec2 x0)Apply If_i_1 with
SNo x2,
SNo_rec_i ((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 (SNo_rec_ii (λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (SNo x5) (SNo_rec_i ((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))) x3 x4) x5) 0))) x2,
0,
λ x3 x4 . x4 = x0 x1 x2 (SNo_rec2 x0) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply SNo_rec2_eq_1 with
x0,
x1,
SNo_rec_ii (λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (SNo x5) (SNo_rec_i ((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))) x3 x4) x5) 0),
x2,
λ x3 x4 . x4 = x0 x1 x2 (SNo_rec_ii (λ x5 . λ x6 : ι → ι → ι . λ x7 . If_i (SNo x7) (SNo_rec_i ... ...) 0)) leaving 4 subgoals.
Apply SNo_rec_ii_eq with
λ x3 . λ x4 : ι → ι → ι . λ x5 . If_i (SNo x5) (SNo_rec_i ((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))) x3 x4) x5) 0,
x1,
λ x3 x4 : ι → ι . x4 x2 = x0 x1 x2 (SNo_rec2 x0) leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H1.
The subproof is completed by applying L4.