Let x0 of type ι be given.
Let x1 of type ι be given.
Apply omega_SNo with
x0.
The subproof is completed by applying H0.
Apply omega_SNo with
x1.
The subproof is completed by applying H1.
Apply mul_SNo_oneL with
x0,
λ x2 x3 . x2 = x1 leaving 2 subgoals.
The subproof is completed by applying L3.
Apply mul_SNo_oneL with
x1,
λ x2 x3 . mul_SNo 1 x0 = x2 leaving 2 subgoals.
The subproof is completed by applying L4.
Apply eps_1_half_eq3 with
λ x2 x3 . mul_SNo x2 x0 = mul_SNo x2 x1.
Apply mul_SNo_assoc with
eps_ 1,
2,
x0,
λ x2 x3 . x2 = mul_SNo (mul_SNo (eps_ 1) 2) x1 leaving 4 subgoals.
The subproof is completed by applying SNo_eps_1.
The subproof is completed by applying SNo_2.
The subproof is completed by applying L3.
Apply mul_SNo_assoc with
eps_ 1,
2,
x1,
λ x2 x3 . mul_SNo (eps_ 1) (mul_SNo 2 x0) = x2 leaving 4 subgoals.
The subproof is completed by applying SNo_eps_1.
The subproof is completed by applying SNo_2.
The subproof is completed by applying L4.
Claim L5: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2
Let x4 of type ι → ο be given.
set y5 to be λ x5 . x4
Apply mul_nat_mul_SNo with
2,
y2,
λ x6 x7 . x6 = mul_SNo 2 y3,
λ x6 x7 . y5 (mul_SNo (eps_ 1) x6) (mul_SNo (eps_ 1) x7) leaving 4 subgoals.
Apply nat_p_omega with
2.
The subproof is completed by applying nat_2.
The subproof is completed by applying H0.
Apply mul_nat_mul_SNo with
2,
y3,
λ x6 x7 . mul_nat 2 y2 = x6 leaving 3 subgoals.
Apply nat_p_omega with
2.
The subproof is completed by applying nat_2.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Let x4 of type ι → ι → ο be given.
Apply L5 with
λ x5 . x4 x5 y3 ⟶ x4 y3 x5.
Assume H6: x4 y3 y3.
The subproof is completed by applying H6.