Let x0 of type ι be given.
Apply H0 with
x0 ∈ SNoS_ omega.
Let x1 of type ι be given.
Apply H1 with
x0 ∈ SNoS_ omega.
Apply H3 with
x0 ∈ SNoS_ omega.
Let x2 of type ι be given.
Apply H4 with
x0 ∈ SNoS_ omega.
Apply H6 with
x0 ∈ SNoS_ omega leaving 2 subgoals.
Apply H7 with
λ x3 x4 . x4 ∈ SNoS_ omega.
Apply nonneg_diadic_rational_p_SNoS_omega with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply omega_nat_p with
x2.
The subproof is completed by applying H5.
Apply H7 with
λ x3 x4 . x4 ∈ SNoS_ omega.
Apply minus_SNo_SNoS_omega with
mul_SNo (eps_ x1) x2.
Apply nonneg_diadic_rational_p_SNoS_omega with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply omega_nat_p with
x2.
The subproof is completed by applying H5.