Apply sqrt_SNo_nonneg_eq with
1,
λ x0 x1 . x1 = 1 leaving 2 subgoals.
The subproof is completed by applying SNo_1.
Apply set_ext with
famunion omega (λ x0 . (λ x1 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 0) x0),
1,
λ x0 x1 . SNoCut x1 (famunion omega (λ x2 . (λ x3 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x3) 1) x2)) = 1 leaving 3 subgoals.
Let x0 of type ι be given.
Apply famunionE_impred with
omega,
λ x1 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 0,
x0,
x0 ∈ 1 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Apply L0 with
x1,
x0 ∈ ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 0 ⟶ x0 ∈ 1 leaving 2 subgoals.
Apply omega_nat_p with
x1.
The subproof is completed by applying H2.
Apply H3 with
λ x2 x3 . x0 ∈ x3 ⟶ x0 ∈ 1.
Assume H5: x0 ∈ 1.
The subproof is completed by applying H5.
Let x0 of type ι be given.
Assume H1: x0 ∈ 1.
Apply cases_1 with
x0,
λ x1 . x1 ∈ famunion omega (λ x2 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x2) 0) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply famunionI with
omega,
λ x1 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 0,
0,
0 leaving 2 subgoals.
Apply nat_p_omega with
0.
The subproof is completed by applying nat_0.
Apply L0 with
0,
0 ∈ (λ x1 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 0) 0 leaving 2 subgoals.
The subproof is completed by applying nat_0.
Apply H2 with
λ x1 x2 . 0 ∈ x2.
The subproof is completed by applying In_0_1.
Apply Empty_eq with
famunion omega (λ x0 . (λ x1 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 1) x0),
λ x0 x1 . SNoCut 1 x1 = 1 leaving 2 subgoals.
Let x0 of type ι be given.
Apply famunionE_impred with
omega,
λ x1 . ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 1,
x0,
False leaving 2 subgoals.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Apply L0 with
x1,
x0 ∈ ap (SNo_sqrtaux 1 sqrt_SNo_nonneg x1) 1 ⟶ False leaving 2 subgoals.
Apply omega_nat_p with
x1.
The subproof is completed by applying H2.
Apply H4 with
λ x2 x3 . nIn x0 x3.
The subproof is completed by applying EmptyE with x0.
Apply SNoL_1 with
λ x0 x1 . SNoCut x0 0 = 1.
Apply SNoR_1 with
λ x0 x1 . SNoCut (SNoL 1) ... = 1.