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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: even_nat x0.
Apply H0 with odd_nat (ordsucc x0).
Assume H1: x0 ∈ omega.
Assume H2: ∃ x1 . and (x1 ∈ omega) (x0 = mul_nat 2 x1).
Apply exactly1of2_E with even_nat x0, even_nat (ordsucc x0), odd_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.
Assume H3: even_nat x0.
Assume H4: not (even_nat (ordsucc x0)).
Apply even_nat_or_odd_nat with ordsucc x0, odd_nat (ordsucc x0) leaving 3 subgoals.
Apply nat_ordsucc with x0.
Apply omega_nat_p with x0.
The subproof is completed by applying H1.
Assume H5: even_nat (ordsucc x0).
Apply FalseE with odd_nat (ordsucc x0).
Apply H4.
The subproof is completed by applying H5.
Assume H5: odd_nat (ordsucc x0).
The subproof is completed by applying H5.
Assume H3: not (even_nat x0).
Apply FalseE with even_nat (ordsucc x0) ⟶ odd_nat (ordsucc x0).
Apply H3.
The subproof is completed by applying H0.
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