Let x0 of type ι → ι be given.

Assume H0: ∀ x1 . 1eb0a.. x1 ⟶ and (SNo (x0 x1)) (∃ x2 . and (SNo x2) (∃ x3 . and (SNo x3) (∃ x4 . and (SNo x4) (∃ x5 . and (SNo x5) (∃ x6 . and (SNo x6) (∃ x7 . and (SNo x7) (∃ x8 . and (SNo x8) (x1 = bbc71.. (x0 x1) x2 x3 x4 x5 x6 x7 x8)))))))).

Assume H1: ∀ x1 . 1eb0a.. x1 ⟶ SNo (x0 x1).

Let x1 of type ι → ι be given.

Assume H2: ∀ x2 . 1eb0a.. x2 ⟶ and (SNo (x1 x2)) (∃ x3 . and (SNo x3) (∃ x4 . and (SNo x4) (∃ x5 . and (SNo x5) (∃ x6 . and (SNo x6) (∃ x7 . and (SNo x7) (∃ x8 . and (SNo x8) (x2 = bbc71.. (x0 x2) (x1 x2) x3 x4 x5 x6 x7 x8))))))).

Assume H3: ∀ x2 . 1eb0a.. x2 ⟶ SNo (x1 x2).

Let x2 of type ι → ι be given.

Assume H4: ∀ x3 . 1eb0a.. x3 ⟶ and (SNo (x2 x3)) (∃ x4 . and (SNo x4) (∃ x5 . and (SNo x5) (∃ x6 . and (SNo x6) (∃ x7 . and (SNo x7) (∃ x8 . and (SNo x8) (x3 = bbc71.. (x0 x3) (x1 x3) (x2 x3) x4 x5 x6 x7 x8)))))).

Let x3 of type ι be given.

Assume H5: 1eb0a.. x3.

Apply unknownprop_8229c3616e9e41610bc79f2f56a50fda9fdf97a5060d19a39ffc03f0fce0477f with x0, x1, x2, x3, SNo (8d7df.. x0 x1 x2 x3) 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.

The subproof is completed by applying H5.

Assume H6: SNo (8d7df.. x0 x1 x2 x3).

Assume H7: ∃ x4 . and (SNo x4) (∃ x5 . and (SNo x5) (∃ x6 . and (SNo x6) (∃ x7 . and (SNo x7) (x3 = bbc71.. (x0 x3) (x1 x3) (x2 x3) (8d7df.. x0 x1 x2 x3) x4 x5 x6 x7)))).

The subproof is completed by applying H6.

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Proofgold Proof