Let x0 of type ι be given.
Apply SNoCut_0_0 with
λ x1 x2 . SNoLe x1 (SNoCut (famunion omega (λ x3 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x3) 0)) (famunion omega (λ x3 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x3) 1))).
Apply SNoCut_Le with
0,
0,
famunion omega (λ x1 . (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 0) x1),
famunion omega (λ x1 . (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 1) x1) leaving 4 subgoals.
The subproof is completed by applying SNoCutP_0_0.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Assume H4: x1 ∈ 0.
Let x1 of type ι be given.
Apply SNoCut_0_0 with
λ x2 x3 . SNoLt x3 x1.
Apply H3 with
x1,
SNoLt 0 x1 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply SNoLeE with
0,
x1,
SNoLt 0 x1 leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
Assume H8: 0 = x1.
Apply FalseE with
SNoLt 0 x1.
Apply SNoLt_irref with
x0.
Apply SNoLtLe_tra with
x0,
0,
x0 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H0.
Apply mul_SNo_zeroR with
0,
λ x2 x3 . SNoLt x0 x2 leaving 2 subgoals.
The subproof is completed by applying SNo_0.
Apply H8 with
λ x2 x3 . SNoLt x0 (mul_SNo x3 x3).
The subproof is completed by applying H7.
The subproof is completed by applying H1.