Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Let x3 of type ι be given.

Assume H0: SNoCutP x0 x1.

Assume H1: SNoCutP x2 x3.

Let x4 of type ι be given.

Let x5 of type ι be given.

Assume H2: x4 = SNoCut x0 x1.

Assume H3: x5 = SNoCut x2 x3.

Let x6 of type ι be given.

Assume H4: x6 ∈ SNoR (mul_SNo x4 x5).

Let x7 of type ο be given.

Assume H5: ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x3 ⟶ SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9)) ⟶ x7.

Assume H6: ∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x2 ⟶ SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9)) ⟶ x7.

Apply mul_SNo_SNoCut_SNoR_interpolate with x0, x1, x2, x3, x4, x5, x6, x7 leaving 7 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Assume H7: ∃ x8 . and (x8 ∈ x0) (∃ x9 . and (x9 ∈ x3) (SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9)))).

Apply H7 with x7.

Let x8 of type ι be given.

Assume H8: (λ x9 . and (x9 ∈ x0) (∃ x10 . and (x10 ∈ x3) (SNoLe (add_SNo (mul_SNo x9 x5) (mul_SNo x4 x10)) (add_SNo x6 (mul_SNo x9 x10))))) x8.

Apply H8 with x7.

Assume H9: x8 ∈ x0.

Assume H10: ∃ x9 . and (x9 ∈ x3) (SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9))).

Apply H10 with x7.

Let x9 of type ι be given.

Assume H11: (λ x10 . and (x10 ∈ x3) (SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x10)) (add_SNo x6 (mul_SNo x8 x10)))) x9.

Apply H11 with x7.

Apply H5 with x8, x9.

The subproof is completed by applying H9.

Assume H7: ∃ x8 . and (x8 ∈ x1) (∃ x9 . and (x9 ∈ x2) (SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9)))).

Apply H7 with x7.

Let x8 of type ι be given.

Assume H8: (λ x9 . and (x9 ∈ x1) (∃ x10 . and (x10 ∈ x2) (SNoLe (add_SNo (mul_SNo x9 x5) (mul_SNo x4 x10)) (add_SNo x6 (mul_SNo x9 x10))))) x8.

Apply H8 with x7.

Assume H9: x8 ∈ x1.

Assume H10: ∃ x9 . and (x9 ∈ x2) (SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9))).

Apply H10 with x7.

Let x9 of type ι be given.

Assume H11: (λ x10 . and (x10 ∈ x2) (SNoLe ... ...)) ....

...

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Proofgold Proof