Let x0 of type ι be given.
Let x1 of type ο be given.
Apply H1 with
0.
Apply andI with
0 ∈ omega,
∃ x2 . and (x2 ∈ omega) (or (x0 = mul_SNo (eps_ 0) x2) (x0 = minus_SNo (mul_SNo (eps_ 0) x2))) leaving 2 subgoals.
Apply nat_p_omega with
0.
The subproof is completed by applying nat_0.
Let x2 of type ο be given.
Apply H2 with
x0.
Apply andI with
x0 ∈ omega,
or (x0 = mul_SNo (eps_ 0) x0) (x0 = minus_SNo (mul_SNo (eps_ 0) x0)) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply orIL with
x0 = mul_SNo (eps_ 0) x0,
x0 = minus_SNo (mul_SNo (eps_ 0) x0).
Apply eps_0_1 with
λ x3 x4 . x0 = mul_SNo x4 x0.
Let x3 of type ι → ι → ο be given.
Apply mul_SNo_oneL with
x0,
λ x4 x5 . x3 x5 x4.
Apply omega_SNo with
x0.
The subproof is completed by applying H0.