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
λ x3 x4 . x4 ∈ SNoS_ omega.
Apply int_SNo_cases with
λ x3 . mul_SNo (eps_ x1) x3 ∈ SNoS_ omega,
x2 leaving 3 subgoals.
Let x3 of type ι be given.
Apply nonneg_diadic_rational_p_SNoS_omega with
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply omega_nat_p with
x3.
The subproof is completed by applying H7.
Let x3 of type ι be given.
Apply mul_SNo_minus_distrR with
eps_ x1,
x3,
λ x4 x5 . x5 ∈ SNoS_ omega leaving 3 subgoals.
Apply SNo_eps_ with
x1.
The subproof is completed by applying H2.
Apply omega_SNo with
x3.
The subproof is completed by applying H7.
Apply minus_SNo_SNoS_omega with
mul_SNo (eps_ x1) x3.
Apply nonneg_diadic_rational_p_SNoS_omega with
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply omega_nat_p with
x3.
The subproof is completed by applying H7.
The subproof is completed by applying H5.