Let x0 of type ι → ο be given.
Assume H0:
∀ x1 . ordinal x1 ⟶ (∀ x2 . prim1 x2 x1 ⟶ x0 x2) ⟶ x0 x1.
Apply In_ind with
λ x1 . ordinal x1 ⟶ x0 x1.
Let x1 of type ι be given.
Apply H0 with
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Apply H1 with
x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply ordinal_Hered with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.