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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Apply add_SNo_assoc with x0, minus_SNo x1, add_SNo x1 x2, λ x3 x4 . x3 = add_SNo x0 x2 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with x1.
The subproof is completed by applying H1.
Apply SNo_add_SNo with x1, x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
set y3 to be add_SNo x0 x2
Claim L3: ∀ x4 : ι → ο . x4 y3x4 (add_SNo x0 (add_SNo (minus_SNo x1) (add_SNo x1 x2)))
Let x4 of type ιο be given.
Apply add_SNo_minus_L2 with x2, y3, λ x5 x6 . (λ x7 . x4) (add_SNo x1 x5) (add_SNo x1 x6) leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ιιο be given.
Apply L3 with λ x5 . x4 x5 y3x4 y3 x5.
Assume H4: x4 y3 y3.
The subproof is completed by applying H4.