Let x0 of type ι be given.
Apply SNoS_omega_diadic_rational_p with
x0,
x0 ∈ rational leaving 2 subgoals.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Apply H1 with
x0 ∈ rational.
Apply H3 with
x0 ∈ rational.
Let x2 of type ι be given.
Apply H4 with
x0 ∈ rational.
Apply omega_nat_p with
x1.
The subproof is completed by applying H2.
Apply SepI with
real,
λ x3 . ∃ x4 . and (x4 ∈ int) (∃ x5 . and (x5 ∈ setminus omega (Sing 0)) (x3 = div_SNo x4 x5)),
x0 leaving 2 subgoals.
Apply SNoS_omega_real with
x0.
The subproof is completed by applying H0.
Let x3 of type ο be given.
Apply H8 with
x2.
Apply andI with
x2 ∈ int,
∃ x4 . and (x4 ∈ setminus omega (Sing 0)) (x0 = div_SNo x2 x4) leaving 2 subgoals.
The subproof is completed by applying H5.
Let x4 of type ο be given.
Apply H9 with
exp_SNo_nat 2 x1.
Apply andI with
exp_SNo_nat 2 x1 ∈ setminus omega (Sing 0),
x0 = div_SNo x2 (exp_SNo_nat 2 x1) leaving 2 subgoals.
Apply setminusI with
omega,
Sing 0,
exp_SNo_nat 2 x1 leaving 2 subgoals.
Apply nat_p_omega with
exp_SNo_nat 2 x1.
Apply nat_exp_SNo_nat with
2,
x1 leaving 2 subgoals.
The subproof is completed by applying nat_2.
The subproof is completed by applying L7.
Apply neq_1_0.
Apply mul_SNo_eps_power_2 with
x1,
λ x5 x6 . x5 = 0 leaving 2 subgoals.
The subproof is completed by applying L7.
Apply SingE with
0,
exp_SNo_nat 2 x1,
λ x5 x6 . mul_SNo (eps_ x1) x6 = 0 leaving 2 subgoals.
The subproof is completed by applying H10.
Apply mul_SNo_zeroR with
eps_ x1.
Apply SNo_eps_ with
x1.
The subproof is completed by applying H2.
Apply H6 with
λ x5 x6 . x6 = mul_SNo x2 (recip_SNo (exp_SNo_nat 2 x1)).
Apply recip_SNo_pow_2 with
x1,
λ x5 x6 . mul_SNo (eps_ x1) x2 = mul_SNo x2 x6 leaving 2 subgoals.
The subproof is completed by applying L7.
Apply mul_SNo_com with
eps_ x1,
x2 leaving 2 subgoals.
Apply SNo_eps_ with
x1.
The subproof is completed by applying H2.
Apply int_SNo with
x2.
The subproof is completed by applying H5.