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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: nat_p x0.
Claim L1: SNo (eps_ x0)
Apply SNo_eps_ with x0.
Apply nat_p_omega with x0.
The subproof is completed by applying H0.
Claim L2: SNoLt 0 (eps_ x0)
Apply SNo_eps_pos with x0.
Apply nat_p_omega with x0.
The subproof is completed by applying H0.
Apply mul_SNo_nonzero_cancel with eps_ x0, recip_SNo_pos (eps_ x0), exp_SNo_nat 2 x0 leaving 5 subgoals.
The subproof is completed by applying L1.
Assume H3: eps_ x0 = 0.
Apply SNoLt_irref with 0.
Apply H3 with λ x1 x2 . SNoLt 0 x1.
The subproof is completed by applying L2.
Apply SNo_recip_SNo_pos with eps_ x0 leaving 2 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L2.
Apply SNo_exp_SNo_nat with 2, x0 leaving 2 subgoals.
The subproof is completed by applying SNo_2.
The subproof is completed by applying H0.
Apply mul_SNo_eps_power_2 with x0, λ x1 x2 . mul_SNo (eps_ x0) (recip_SNo_pos (eps_ x0)) = x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply recip_SNo_pos_invR with eps_ x0 leaving 2 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L2.