Let x0 of type ι be given.
Apply H1 with
even_nat x0.
Apply andI with
x0 ∈ omega,
∃ x1 . and (x1 ∈ omega) (x0 = mul_nat 2 x1) leaving 2 subgoals.
Apply nat_p_omega with
x0.
The subproof is completed by applying H0.
Apply H3 with
∃ x1 . and (x1 ∈ omega) (x0 = mul_nat 2 x1).
Let x1 of type ι be given.
Apply H4 with
∃ x2 . and (x2 ∈ omega) (x0 = mul_nat 2 x2).
Apply nat_inv with
x1,
∃ x2 . and (x2 ∈ omega) (x0 = mul_nat 2 x2) leaving 3 subgoals.
Apply omega_nat_p with
x1.
The subproof is completed by applying H5.
Assume H7: x1 = 0.
Apply neq_ordsucc_0 with
ordsucc x0,
∃ x2 . and (x2 ∈ omega) (x0 = mul_nat 2 x2).
set y3 to be 0
Claim L8: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2
Let x4 of type ι → ο be given.
Assume H8: x4 0.
Apply H6 with
λ x5 . x4.
Apply H7 with
λ x5 x6 . mul_nat 2 x6 = 0,
λ x5 . x4 leaving 2 subgoals.
The subproof is completed by applying mul_nat_0R with 2.
The subproof is completed by applying H8.
Let x4 of type ι → ι → ο be given.
Apply L8 with
λ x5 . x4 x5 y3 ⟶ x4 y3 x5.
Assume H9: x4 y3 y3.
The subproof is completed by applying H9.
Apply H7 with
∃ x2 . and (x2 ∈ omega) (x0 = mul_nat 2 x2).
Let x2 of type ι be given.
Apply H8 with
∃ x3 . and (x3 ∈ omega) (x0 = mul_nat 2 x3).
Let x3 of type ο be given.
Apply H11 with
x2.
Apply andI with
x2 ∈ omega,
x0 = mul_nat 2 x2 leaving 2 subgoals.
Apply nat_p_omega with
x2.
The subproof is completed by applying H9.
Apply ordsucc_inj with
x0,
mul_nat 2 x2.