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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: odd_nat x0.
Apply H0 with even_nat (ordsucc x0).
Assume H1: x0omega.
Assume H2: ∀ x1 . x1omegax0 = 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.
Assume H3: even_nat x0.
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.
Assume H3: not (even_nat x0).
Assume H4: even_nat (ordsucc x0).
The subproof is completed by applying H4.