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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: nat_p x0.
set y1 to be 1
Claim L1: ∀ x2 : ι → ο . x2 y1x2 (mul_SNo (exp_SNo_nat 2 x0) (eps_ x0))
Let x2 of type ιο be given.
Assume H1: x2 1.
Apply mul_SNo_com with exp_SNo_nat 2 y1, eps_ y1, λ x3 . x2 leaving 3 subgoals.
Apply SNo_exp_SNo_nat with 2, y1 leaving 2 subgoals.
The subproof is completed by applying SNo_2.
The subproof is completed by applying H0.
Apply SNo_eps_ with y1.
Apply nat_p_omega with y1.
The subproof is completed by applying H0.
Apply mul_SNo_eps_power_2 with y1, λ x3 . x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x2 of type ιιο be given.
Apply L1 with λ x3 . x2 x3 y1x2 y1 x3.
Assume H2: x2 y1 y1.
The subproof is completed by applying H2.