Let x0 of type ι be given.

Let x1 of type ι → ι → ο be given.

Assume H0: ∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2.

Assume H1: ∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4).

Assume H2: ∀ x2 . x2 ⊆ x0 ⟶ atleastp u6 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4)).

Let x2 of type ι be given.

Let x3 of type ι be given.

Let x4 of type ι be given.

Let x5 of type ι be given.

Assume H3: x2 ⊆ x0.

Assume H4: x3 ⊆ x0.

Assume H5: x4 ⊆ x0.

Assume H6: x5 ⊆ x0.

Assume H7: ∀ x6 . x6 ∈ x2 ⟶ nIn x6 x5.

Assume H8: ∀ x6 . x6 ∈ x2 ⟶ nIn x6 x3.

Assume H9: ∀ x6 . x6 ∈ x4 ⟶ nIn x6 x2.

Assume H10: ∀ x6 . x6 ∈ x4 ⟶ nIn x6 x3.

Assume H11: ∀ x6 . x6 ∈ x4 ⟶ nIn x6 x5.

Assume H12: ∀ x6 . x6 ∈ x3 ⟶ nIn x6 x5.

Let x6 of type ι be given.

Let x7 of type ι be given.

Let x8 of type ι be given.

Let x9 of type ι be given.

Let x10 of type ι be given.

Assume H13: x4 = SetAdjoin (SetAdjoin (UPair x6 x7) x8) x9.

Assume H14: x10 ∈ x5.

Assume H15: x7 = x6 ⟶ ∀ x11 : ο . x11.

Assume H16: x8 = x6 ⟶ ∀ x11 : ο . x11.

Assume H17: x9 = x6 ⟶ ∀ x11 : ο . x11.

Assume H18: x8 = x7 ⟶ ∀ x11 : ο . x11.

Assume H19: x9 = x7 ⟶ ∀ x11 : ο . x11.

Assume H20: x9 = x8 ⟶ ∀ x11 : ο . x11.

Assume H21: x1 x6 x7.

Assume H22: x1 x7 x8.

Assume H23: x1 x8 x9.

Assume H24: x1 x9 x6.

Assume H25: ∀ x11 . x11 ∈ x4 ⟶ (x11 = x10 ⟶ ∀ x12 : ο . x12) ⟶ not (x1 x11 x10) ⟶ atleastp (binintersect (DirGraphOutNeighbors x0 x1 x11) (DirGraphOutNeighbors x0 x1 x10)) u2.

Assume H28: not (x1 x7 x10).

Assume H29: not (x1 x9 x10).

Let x11 of type ι → ι be given.

Let x12 of type ι → ι be given.

Assume H30: ∀ x13 . x13 ∈ x4 ⟶ x11 x13 ∈ x2.

Assume H31: ∀ x13 . x13 ∈ x4 ⟶ x11 x13 ∈ DirGraphOutNeighbors x0 x1 x13.

Assume H32: ∀ x13 . x13 ∈ x4 ⟶ x12 x13 ∈ x3.

...

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Proofgold Proof