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.

Assume H2: nat_p x2.

Assume H3: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4) ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x4)) x2) (equip (binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x4)) (ordsucc x2)).

Let x3 of type ι be given.

Let x4 of type ι be given.

Assume H4: x3 ∈ x0.

Assume H5: x4 ∈ DirGraphOutNeighbors x0 x1 x3.

Assume H6: ∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ not (x1 x4 x5).

Let x5 of type ι → ι → ο be given.

Apply unknownprop_2ec38089a92dcd86a75a337a6e999322444786992f1b612acfe1d68011bad142 with {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) (ordsucc x2)}, setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3), λ x6 x7 . x5 x7 x6.

Let x6 of type ι be given.

Assume H7: x6 ∈ setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3).

Apply setminusE with DirGraphOutNeighbors x0 x1 x4, Sing x3, x6, x6 ∈ {x7 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x3)) (ordsucc x2)} leaving 2 subgoals.

The subproof is completed by applying H7.

Assume H8: x6 ∈ DirGraphOutNeighbors x0 x1 x4.

Assume H9: nIn x6 ....

...

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Proofgold Proof