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Proofgold Proof

pf
Apply add_SNo_cancel_L with 0, minus_SNo 0, 0 leaving 4 subgoals.
The subproof is completed by applying SNo_0.
Apply SNo_minus_SNo with 0.
The subproof is completed by applying SNo_0.
The subproof is completed by applying SNo_0.
set y0 to be add_SNo 0 (minus_SNo 0)
set y1 to be add_SNo 0 0
Claim L0: ∀ x2 : ι → ο . x2 y1x2 y0
Let x2 of type ιο be given.
Assume H0: x2 (add_SNo 0 0).
Apply add_SNo_minus_SNo_rinv with 0, λ x3 . x2 leaving 2 subgoals.
The subproof is completed by applying SNo_0.
set y3 to be λ x3 . x2
Apply add_SNo_0L with 0, λ x4 x5 . y3 x5 x4 leaving 2 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H0.
Let x2 of type ιιο be given.
Apply L0 with λ x3 . x2 x3 y1x2 y1 x3.
Assume H1: x2 y1 y1.
The subproof is completed by applying H1.