Let x0 of type ι be given.
Apply H0 with
x0 ∈ setminus omega 2.
Apply H1 with
(∀ x1 . x1 ∈ omega ⟶ divides_nat x1 x0 ⟶ or (x1 = 1) (x1 = x0)) ⟶ x0 ∈ setminus omega 2.
Assume H3: 1 ∈ x0.
Apply setminusI with
omega,
2,
x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H5: x0 ∈ 2.
Apply cases_2 with
x0,
λ x1 . x0 = x1 ⟶ False leaving 4 subgoals.
The subproof is completed by applying H5.
Assume H6: x0 = 0.
Apply In_no2cycle with
0,
1 leaving 2 subgoals.
The subproof is completed by applying In_0_1.
Apply H6 with
λ x1 x2 . 1 ∈ x1.
The subproof is completed by applying H3.
Assume H6: x0 = 1.
Apply In_irref with
1.
Apply H6 with
λ x1 x2 . 1 ∈ x1.
The subproof is completed by applying H3.
Let x1 of type ι → ι → ο be given.
Assume H6: x1 x0 x0.
The subproof is completed by applying H6.