Let x0 of type ι be given.
Apply H0 with
even_nat (ordsucc x0).
Assume H1:
x0 ∈ omega.
Assume H2:
∀ x1 . x1 ∈ omega ⟶ x0 = mul_nat 2 x1 ⟶ ∀ x2 : ο . x2.
Apply exactly1of2_E with
even_nat x0,
even_nat (ordsucc x0),
even_nat (ordsucc x0) leaving 3 subgoals.
Apply even_nat_xor_S with
x0.
Apply omega_nat_p with
x0.
The subproof is completed by applying H1.
Apply FalseE with
not (even_nat (ordsucc x0)) ⟶ even_nat (ordsucc x0).
Apply even_nat_not_odd_nat with
x0 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H0.
The subproof is completed by applying H4.