Let x0 of type ι → ο be given.
Assume H0: x0 0.
Claim L3:
∀ x1 . x1 ∈ omega ⟶ x0 x1Let x1 of type ι be given.
Apply xm with
x1 = 0,
x0 x1 leaving 2 subgoals.
Assume H4: x1 = 0.
Apply H4 with
λ x2 x3 . x0 x3.
The subproof is completed by applying H0.
Assume H4: x1 = 0 ⟶ ∀ x2 : ο . x2.
Apply H1 with
x1.
Apply setminusI with
omega,
1,
x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H5: x1 ∈ 1.
Apply H4.
Apply cases_1 with
x1,
λ x2 . x2 = 0 leaving 2 subgoals.
The subproof is completed by applying H5.
Let x2 of type ι → ι → ο be given.
Assume H6: x2 0 0.
The subproof is completed by applying H6.
Let x1 of type ι be given.
Apply xm with
x1 = 0,
x0 (minus_SNo x1) leaving 2 subgoals.
Assume H5: x1 = 0.
Apply H5 with
λ x2 x3 . x0 (minus_SNo x3).
Apply minus_SNo_0 with
λ x2 x3 . x0 x3.
The subproof is completed by applying H0.
Assume H5: x1 = 0 ⟶ ∀ x2 : ο . x2.
Apply H2 with
x1.
Apply setminusI with
omega,
1,
x1 leaving 2 subgoals.
The subproof is completed by applying H4.
Assume H6: x1 ∈ 1.
Apply H5.
Apply cases_1 with
x1,
λ x2 . x2 = 0 leaving 2 subgoals.
The subproof is completed by applying H6.
Let x2 of type ι → ι → ο be given.
Assume H7: x2 0 0.
The subproof is completed by applying H7.
Apply unknownprop_2504c05a08587fe0873ed45685efc417625f0a904281d653d757d643896f9a70 with
x0 leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L4.