Let x0 of type ι → ι be given.
Apply nat_ind with
λ x1 . (∀ x2 . x2 ∈ x1 ⟶ nat_p (x0 x2)) ⟶ ∀ x2 . x2 ∈ x1 ⟶ divides_nat (x0 x2) (Pi_nat x0 x1) leaving 2 subgoals.
Assume H0:
∀ x1 . x1 ∈ 0 ⟶ nat_p (x0 x1).
Let x1 of type ι be given.
Assume H1: x1 ∈ 0.
Apply FalseE with
divides_nat (x0 x1) (Pi_nat x0 0).
Apply EmptyE with
x1.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Claim L3:
∀ x2 . x2 ∈ x1 ⟶ nat_p (x0 x2)Let x2 of type ι be given.
Assume H3: x2 ∈ x1.
Apply H2 with
x2.
Apply ordsuccI1 with
x1,
x2.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Apply Pi_nat_S with
x0,
x1,
λ x3 x4 . divides_nat (x0 x2) x4 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply ordsuccE with
x1,
x2,
divides_nat (x0 x2) (mul_nat (Pi_nat x0 x1) (x0 x1)) leaving 3 subgoals.
The subproof is completed by applying H4.
Assume H5: x2 ∈ x1.
Apply unknownprop_f799b99d854d7bca6941dc7751c0c00a5bf29ac2d7e070aa318a7a7ed9ce8fa0 with
x0 x2,
Pi_nat x0 x1,
mul_nat (Pi_nat x0 x1) (x0 x1) leaving 2 subgoals.
Apply H1 with
x2 leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H5.
Apply divides_nat_mulR with
Pi_nat x0 x1,
x0 x1 leaving 2 subgoals.
Apply nat_p_omega with
Pi_nat x0 x1.
Apply Pi_nat_p with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L3.
Apply nat_p_omega with
x0 x1.
Apply H2 with
x1.
The subproof is completed by applying ordsuccI2 with x1.
Assume H5: x2 = x1.
Apply H5 with
λ x3 x4 . divides_nat (x0 x4) (mul_nat (Pi_nat x0 x1) (x0 x1)).
Apply divides_nat_mulL with
Pi_nat x0 x1,
x0 x1 leaving 2 subgoals.
Apply nat_p_omega with
Pi_nat x0 x1.
Apply Pi_nat_p with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L3.
Apply nat_p_omega with
x0 x1.
Apply H2 with
x1.
The subproof is completed by applying ordsuccI2 with x1.