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Proofgold Proof

pf
Apply SNoLev_ind with λ x0 . mul_SNo x0 1 = x0.
Let x0 of type ι be given.
Assume H0: SNo x0.
Assume H1: ∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ mul_SNo x1 1 = x1.
Apply mul_SNo_eq_3 with x0, 1, mul_SNo x0 1 = x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_1.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H2: SNoCutP x1 x2.
Assume H3: ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoL 1 ⟶ x3 = add_SNo (mul_SNo x5 1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoR 1 ⟶ x3 = add_SNo (mul_SNo x5 1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ⟶ x4) ⟶ x4.
Assume H4: ∀ x3 . x3 ∈ SNoL x0 ⟶ ∀ x4 . x4 ∈ SNoL 1 ⟶ add_SNo (mul_SNo x3 1) (add_SNo (mul_SNo x0 x4) (minus_SNo (mul_SNo x3 x4))) ∈ x1.
Assume H5: ∀ x3 . x3 ∈ SNoR x0 ⟶ ∀ x4 . x4 ∈ SNoR 1 ⟶ add_SNo (mul_SNo x3 1) (add_SNo (mul_SNo x0 x4) (minus_SNo (mul_SNo x3 x4))) ∈ x1.
Assume H6: ∀ x3 . x3 ∈ x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoR 1 ⟶ x3 = add_SNo (mul_SNo x5 1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoL 1 ⟶ x3 = add_SNo (mul_SNo x5 1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ⟶ x4) ⟶ x4.
Assume H7: ∀ x3 . x3 ∈ SNoL x0 ⟶ ∀ x4 . x4 ∈ SNoR 1 ⟶ add_SNo (mul_SNo x3 1) (add_SNo (mul_SNo x0 x4) (minus_SNo (mul_SNo x3 x4))) ∈ x2.
Assume H8: ∀ x3 . x3 ∈ SNoR x0 ⟶ ∀ x4 . x4 ∈ SNoL 1 ⟶ add_SNo (mul_SNo x3 1) (add_SNo (mul_SNo x0 x4) (minus_SNo (mul_SNo x3 x4))) ∈ x2.
Assume H9: mul_SNo x0 1 = SNoCut x1 x2.
Apply mul_SNo_prop_1 with x0, 1, mul_SNo x0 1 = x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_1.
Assume H10: SNo (mul_SNo x0 1).
Assume H11: ∀ x3 . ... ⟶ ∀ x4 . ... ⟶ SNoLt (add_SNo (mul_SNo ... 1) ...) ....
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