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).

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.

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.

Let x11 of type ι be given.

Assume H2: x9 ⊆ x0.

Assume H3: x11 ⊆ x0.

Assume H4: x8 = setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x5).

Assume H5: x10 = setminus (DirGraphOutNeighbors x0 x1 x5) (Sing x4).

Assume H6: x9 = {x12 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4))|equip (binintersect (DirGraphOutNeighbors x0 x1 x12) (DirGraphOutNeighbors x0 x1 x4)) x2}.

Assume H7: x11 = setminus {x12 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4))|equip (binintersect (DirGraphOutNeighbors x0 x1 x12) (DirGraphOutNeighbors x0 x1 x4)) x3} x10.

Assume H8: ∀ x12 . x12 ∈ x9 ⟶ nIn x12 x8.

Assume H9: ∀ x12 . x12 ∈ x9 ⟶ nIn x12 x11.

Assume H10: ∀ x12 . x12 ∈ x8 ⟶ nIn x12 x11.

Assume H11: x6 ∈ x9.

Assume H12: x7 ∈ x11.

Assume H13: x1 x6 x7.

Let x12 of type ι → ι be given.

Let x13 of type ι → ι be given.

Assume H14: x1 x6 (x12 x6).

Assume H15: ∀ x14 . x14 ∈ x8 ⟶ x13 x14 ∈ {x15 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4))|equip (binintersect (DirGraphOutNeighbors x0 x1 x15) (DirGraphOutNeighbors x0 x1 x4)) x2}.

Assume H16: ∀ x14 . x14 ∈ x8 ⟶ x12 (x13 x14) = x14.

Claim L17: ...

...

Claim L18: ...

...

Claim L19: ...

...

Claim L20: ...

...

Claim L21: ...

...

Apply L21 with atleastp x3 {x14 ∈ setminus x9 (Sing x6)|x1 (x12 x14) x7}.

Let x14 of type ι → ι be given.

Assume H22: inj x3 (binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x4)) x14.

Apply H22 with atleastp x3 {x15 ∈ setminus x9 (Sing x6)|x1 (x12 x15) x7}.

Assume H23: ∀ x15 . x15 ∈ x3 ⟶ x14 x15 ∈ binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x4).

Assume H24: ∀ x15 . x15 ∈ x3 ⟶ ∀ x16 . x16 ∈ x3 ⟶ x14 x15 = x14 x16 ⟶ x15 = x16.

Let x15 of type ο be given.

Assume H25: ∀ x16 : ι → ι . inj x3 {x17 ∈ setminus x9 (Sing ...)|...} ... ⟶ x15.

...

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Proofgold Proof