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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Apply add_SNo_prop1 with x0, x1, SNo (add_SNo x0 x1) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H2: and (and (and (and (SNo (add_SNo x0 x1)) (∀ x2 . x2SNoL x0SNoLt (add_SNo x2 x1) (add_SNo x0 x1))) (∀ x2 . x2SNoR x0SNoLt (add_SNo x0 x1) (add_SNo x2 x1))) (∀ x2 . x2SNoL x1SNoLt (add_SNo x0 x2) (add_SNo x0 x1))) (∀ x2 . x2SNoR x1SNoLt (add_SNo x0 x1) (add_SNo x0 x2)).
Apply H2 with SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))SNo (add_SNo x0 x1).
Assume H3: and (and (and (SNo (add_SNo x0 x1)) (∀ x2 . x2SNoL x0SNoLt (add_SNo x2 x1) (add_SNo x0 x1))) (∀ x2 . x2SNoR x0SNoLt (add_SNo x0 x1) (add_SNo x2 x1))) (∀ x2 . x2SNoL x1SNoLt (add_SNo x0 x2) (add_SNo x0 x1)).
Apply H3 with (∀ x2 . x2SNoR x1SNoLt (add_SNo x0 x1) (add_SNo x0 x2))SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))SNo (add_SNo x0 x1).
Assume H4: and (and (SNo (add_SNo x0 x1)) (∀ x2 . x2SNoL x0SNoLt (add_SNo x2 x1) (add_SNo x0 x1))) (∀ x2 . x2SNoR x0SNoLt (add_SNo x0 x1) (add_SNo x2 x1)).
Apply H4 with (∀ x2 . ...SNoLt (add_SNo x0 x2) ...)(∀ x2 . x2SNoR x1SNoLt (add_SNo x0 x1) (add_SNo x0 x2))SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))SNo (add_SNo x0 x1).
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