Apply SNo_rec_i_eq with
λ x0 . λ x1 : ι → ι . SNoCut (famunion omega (λ x2 . ap (SNo_sqrtaux x0 x1 x2) 0)) (famunion omega (λ x2 . ap (SNo_sqrtaux x0 x1 x2) 1)).
Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H1:
∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ x1 x3 = x2 x3.
set y3 to be ...
Let x4 of type ι → ο be given.
Apply famunion_ext with
omega,
λ x5 . ap (SNo_sqrtaux x1 x2 x5) 0,
λ x5 . ap (SNo_sqrtaux x1 y3 x5) 0,
λ x5 x6 . (λ x7 . x4) (SNoCut x5 (famunion omega (λ x7 . ap (SNo_sqrtaux x1 x2 x7) 1))) (SNoCut x6 (famunion omega (λ x7 . ap (SNo_sqrtaux x1 x2 x7) 1))) leaving 2 subgoals.
Let x5 of type ι be given.
Claim L4:
∀ x7 : ι → ο . x7 y6 ⟶ x7 (ap (SNo_sqrtaux x1 x2 x5) 0)Let x7 of type ι → ο be given.
Apply SNo_sqrtaux_ext with
x2,
y3,
x4,
y6,
λ x8 x9 . (λ x10 . x7) (ap x8 0) (ap x9 0) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply omega_nat_p with
y6.
The subproof is completed by applying H3.
Let x7 of type ι → ι → ο be given.
Apply L4 with
λ x8 . x7 x8 y6 ⟶ x7 y6 x8.
Assume H5: x7 y6 y6.
The subproof is completed by applying H5.
Apply famunion_ext with
omega,
λ x5 . ap (SNo_sqrtaux x1 x2 x5) 1,
λ x5 . ap (SNo_sqrtaux x1 y3 x5) 1,
λ x5 x6 . (λ x7 . x4) (SNoCut (famunion omega (λ x7 . ap (SNo_sqrtaux x1 y3 x7) 0)) x5) (SNoCut (famunion omega (λ x7 . ap (SNo_sqrtaux x1 y3 x7) 0)) x6) leaving 2 subgoals.
Let x5 of type ι be given.
set y6 to be ...
Claim L4:
∀ x7 : ι → ο . x7 y6 ⟶ x7 (ap (SNo_sqrtaux x1 x2 x5) 1)Let x7 of type ι → ο be given.
Apply SNo_sqrtaux_ext with
x2,
y3,
x4,
y6,
λ x8 x9 . (λ x10 . x7) ... ... leaving 3 subgoals.
Let x7 of type ι → ι → ο be given.
Apply L4 with
λ x8 . x7 x8 y6 ⟶ x7 y6 x8.
Assume H5: x7 y6 y6.
The subproof is completed by applying H5.
Let x4 of type ι → ι → ο be given.
Apply L2 with
λ x5 . x4 x5 y3 ⟶ x4 y3 x5.
Assume H3: x4 y3 y3.
The subproof is completed by applying H3.