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.

Assume H3: nat_p x2.

Let x4 of type ι be given.

Let x5 of type ι be given.

Assume H4: x5 ∈ DirGraphOutNeighbors x0 x1 x4.

Let x6 of type ι be given.

Let x7 of type ι be given.

Let x8 of type ι be given.

Assume H5: x6 ⊆ x0.

Assume H6: x7 ⊆ x0.

Assume H7: x8 ⊆ x0.

Assume H8: x7 = setminus (DirGraphOutNeighbors x0 x1 x5) (Sing x4).

Assume H9: x8 = setminus {x9 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4))|equip (binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x4)) x3} x7.

Assume H10: equip x6 x2.

Assume H11: ∀ x9 . x9 ∈ x6 ⟶ x9 = x5 ⟶ ∀ x10 : ο . x10.

Assume H12: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 x7.

Assume H13: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 x8.

Assume H14: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x4).

Assume H15: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x5).

Assume H16: ∀ x9 . x9 ∈ x6 ⟶ ∀ x10 . x10 ∈ x6 ⟶ (x9 = x10 ⟶ ∀ x11 : ο . x11) ⟶ ∀ x11 . x11 ∈ binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x10) ⟶ x11 ∈ x6.

Let x9 of type ι → ι be given.

Assume H17: ∀ x10 . x10 ∈ x6 ⟶ x9 x10 ∈ x7.

Assume H18: ∀ x10 . x10 ∈ x6 ⟶ x1 x10 (x9 x10).

Assume H19: ∀ x10 . x10 ∈ x7 ⟶ ∃ x11 . and (x11 ∈ x6) (x9 x11 = x10).

Assume H20: ∀ x10 . x10 ∈ x8 ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) u1) (equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) x3).

Assume H21: equip {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x5) (Sing x5))|equip (binintersect (DirGraphOutNeighbors ... ... ...) ...) ...} ....

...

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Proofgold Proof