Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Apply nat_complete_ind with
λ x4 . ∀ x5 . nat_p x5 ⟶ ∀ x6 . nat_p x6 ⟶ ∀ x7 . nat_p x7 ⟶ ∀ x8 . nat_p x8 ⟶ nat_pair x5 x7 = x4 ⟶ nat_pair x6 x8 = x4 ⟶ x5 = x6,
nat_pair x2 x3,
x0,
x2,
x1,
x3 leaving 8 subgoals.
Let x4 of type ι be given.
Apply nat_inv_impred with
λ x5 . ∀ x6 . nat_p x6 ⟶ ∀ x7 . nat_p x7 ⟶ ∀ x8 . nat_p x8 ⟶ nat_pair x5 x7 = x4 ⟶ nat_pair x6 x8 = x4 ⟶ x5 = x6 leaving 2 subgoals.
Apply nat_inv_impred with
λ x5 . ∀ x6 . nat_p x6 ⟶ ∀ x7 . nat_p x7 ⟶ nat_pair 0 x6 = x4 ⟶ nat_pair x5 x7 = x4 ⟶ 0 = x5 leaving 2 subgoals.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι → ι → ο be given.
Assume H11: x7 0 0.
The subproof is completed by applying H11.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply FalseE with
0 = ordsucc x5.
Apply even_nat_not_odd_nat with
x4 leaving 2 subgoals.
Apply H11 with
λ x8 x9 . even_nat x8.
Apply exp_SNo_nat_S with
2,
x5,
λ x8 x9 . even_nat (mul_SNo x9 (add_SNo (mul_SNo 2 x7) 1)) leaving 3 subgoals.
The subproof is completed by applying SNo_2.
The subproof is completed by applying H7.
Apply mul_SNo_assoc with
2,
exp_SNo_nat 2 x5,
add_SNo (mul_SNo 2 x7) 1,
λ x8 x9 . even_nat x8 leaving 4 subgoals.
The subproof is completed by applying SNo_2.
Apply SNo_exp_SNo_nat with
2,
x5 leaving 2 subgoals.
The subproof is completed by applying SNo_2.
The subproof is completed by applying H7.
Apply SNo_add_SNo with
mul_SNo 2 x7,
1 leaving 2 subgoals.
Apply SNo_mul_SNo with
2,
x7 leaving 2 subgoals.
The subproof is completed by applying SNo_2.
Apply nat_p_SNo with
x7.
The subproof is completed by applying H9.
The subproof is completed by applying SNo_1.