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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.
Assume H3: SNoLt x0 x2.
Apply add_SNo_prop1 with x0, x1, SNoLt (add_SNo x0 x1) (add_SNo x2 x1) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H4: and (and (and (and (SNo (add_SNo x0 x1)) (∀ x3 . x3 ∈ SNoL x0 ⟶ SNoLt (add_SNo x3 x1) (add_SNo x0 x1))) (∀ x3 . x3 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x3 x1))) (∀ x3 . x3 ∈ SNoL x1 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x0 x1))) (∀ x3 . x3 ∈ SNoR x1 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x3)).
Apply H4 with SNoCutP (binunion {add_SNo x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0))) ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1).
Assume H5: and (and (and (SNo (add_SNo x0 x1)) (∀ x3 . x3 ∈ SNoL x0 ⟶ SNoLt (add_SNo x3 x1) (add_SNo x0 x1))) (∀ x3 . x3 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x3 x1))) (∀ x3 . x3 ∈ SNoL x1 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x0 x1)).
Apply H5 with (∀ x3 . x3 ∈ SNoR x1 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x3)) ⟶ SNoCutP (binunion {add_SNo x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0))) ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1).
Assume H6: and (and (SNo (add_SNo x0 x1)) (∀ x3 . x3 ∈ SNoL x0 ⟶ SNoLt (add_SNo x3 x1) (add_SNo x0 x1))) (∀ x3 . x3 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x3 x1)).
Apply H6 with ... ⟶ ... ⟶ SNoCutP (binunion {add_SNo x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) ...)) ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1).
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