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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.
Claim L2: ...
...
Claim L3: ∀ x2 . SNo x2SNoLev x2SNoLev (mul_SNo x0 x1)SNoLt (mul_SNo x0 x1) x2(λ x3 . or ((λ x4 . ∃ x5 . and (x5SNoL x0) (∃ x6 . and (x6SNoR x1) (SNoLe (add_SNo (mul_SNo x5 x1) (mul_SNo x0 x6)) (add_SNo x4 (mul_SNo x5 x6))))) x3) ((λ x4 . ∃ x5 . and (x5SNoR x0) (∃ x6 . and (x6SNoL x1) (SNoLe (add_SNo (mul_SNo x5 x1) (mul_SNo x0 x6)) (add_SNo x4 (mul_SNo x5 x6))))) x3)) x2
Apply SNoLev_ind with λ x2 . SNoLev x2SNoLev (mul_SNo x0 x1)SNoLt (mul_SNo x0 x1) x2(λ x3 . or ((λ x4 . ∃ x5 . and (x5SNoL x0) (∃ x6 . and (x6SNoR x1) (SNoLe (add_SNo (mul_SNo x5 x1) (mul_SNo x0 x6)) (add_SNo x4 (mul_SNo x5 x6))))) x3) ((λ x4 . ∃ x5 . and (x5SNoR x0) (∃ x6 . and (x6SNoL x1) (SNoLe (add_SNo (mul_SNo x5 x1) (mul_SNo x0 x6)) (add_SNo x4 (mul_SNo x5 x6))))) x3)) x2.
Let x2 of type ι be given.
Assume H3: SNo x2.
Assume H4: ∀ x3 . .........(λ x4 . or ((λ x5 . ∃ x6 . and (x6SNoL x0) (∃ x7 . and (x7SNoR x1) (SNoLe (add_SNo (mul_SNo x6 x1) (mul_SNo x0 x7)) (add_SNo x5 (mul_SNo ... ...))))) ...) ...) ....
...
Let x2 of type ι be given.
Assume H4: x2SNoR (mul_SNo x0 x1).
Apply SNoR_E with mul_SNo x0 x1, x2, or (∃ x3 . and (x3SNoL x0) (∃ x4 . and (x4SNoR x1) (SNoLe (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4)) (add_SNo x2 (mul_SNo x3 x4))))) (∃ x3 . and (x3SNoR x0) (∃ x4 . and (x4SNoL x1) (SNoLe (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4)) (add_SNo x2 (mul_SNo x3 x4))))) leaving 3 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying H4.
Assume H5: SNo x2.
Assume H6: SNoLev x2SNoLev (mul_SNo x0 x1).
Assume H7: SNoLt (mul_SNo x0 x1) x2.
Apply L3 with x2 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.