Let x0 of type ι be given.
Assume H0: x0 ∈ 8.
Let x1 of type ι → ο be given.
Assume H1: x1 0.
Assume H2: x1 1.
Assume H3: x1 2.
Assume H4: x1 3.
Assume H5: x1 4.
Assume H6: x1 5.
Assume H7: x1 6.
Assume H8: x1 7.
Apply ordsuccE with
7,
x0,
x1 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H9: x0 ∈ 7.
Apply cases_7 with
x0,
x1 leaving 8 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
Assume H9: x0 = 7.
Apply H9 with
λ x2 x3 . x1 x3.
The subproof is completed by applying H8.