set y1 to be 1
Claim L0: ∀ x2 : ι → ο . x2 y1 ⟶ x2 y0
Let x2 of type ι → ο be given.
Assume H0: x2 1.
Apply eps_ordsucc_half_add with
0,
λ x3 . x2 leaving 2 subgoals.
The subproof is completed by applying nat_0.
Apply eps_0_1 with
λ x3 . x2.
The subproof is completed by applying H0.
Let x2 of type ι → ι → ο be given.
Apply L0 with
λ x3 . x2 x3 y1 ⟶ x2 y1 x3.
Assume H1: x2 y1 y1.
The subproof is completed by applying H1.