Let x0 of type ι be given.
set y1 to be 1
Let x2 of type ι → ο be given.
Assume H1: x2 1.
Apply mul_SNo_com with
exp_SNo_nat 2 y1,
eps_ y1,
λ x3 . x2 leaving 3 subgoals.
Apply SNo_exp_SNo_nat with
2,
y1 leaving 2 subgoals.
The subproof is completed by applying SNo_2.
The subproof is completed by applying H0.
Apply SNo_eps_ with
y1.
Apply nat_p_omega with
y1.
The subproof is completed by applying H0.
Apply mul_SNo_eps_power_2 with
y1,
λ x3 . x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x2 of type ι → ι → ο be given.
Apply L1 with
λ x3 . x2 x3 y1 ⟶ x2 y1 x3.
Assume H2: x2 y1 y1.
The subproof is completed by applying H2.