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Proofgold Proof

pf
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.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H1: nat_p x2.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H2: 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 H3: x6 ⊆ x0.
Assume H4: x7 ⊆ x0.
Assume H5: x8 ⊆ x0.
Assume H6: x7 = setminus (DirGraphOutNeighbors x0 x1 x5) (Sing x4).
Assume H7: 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 H8: equip x6 x2.
Assume H9: ∀ x9 . x9 ∈ x6 ⟶ x9 = x5 ⟶ ∀ x10 : ο . x10.
Assume H10: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 x7.
Assume H11: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 x8.
Assume H12: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x4).
Assume H13: ∀ x9 . x9 ∈ x6 ⟶ nIn x9 (DirGraphOutNeighbors x0 x1 x5).
Assume H14: ∀ 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 H15: ∀ x10 . x10 ∈ x6 ⟶ x9 x10 ∈ x7.
Assume H16: ∀ x10 . x10 ∈ x6 ⟶ x1 x10 (x9 x10).
Assume H17: ∀ x10 . x10 ∈ x7 ⟶ ∃ x11 . and (x11 ∈ x6) (x9 x11 = x10).
Assume H18: ∀ 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 H19: equip {x10 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x5) (Sing x5))|equip (binintersect (DirGraphOutNeighbors x0 x1 x10) (DirGraphOutNeighbors x0 x1 x5)) u1} x2.
Let x10 of type ι be given.
Assume H20: x10 ∈ x8.
Claim L21: ...
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Apply setminusE with {x11 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x4) (Sing x4))|equip (binintersect (DirGraphOutNeighbors x0 x1 x11) (DirGraphOutNeighbors x0 x1 x4)) x3}, x7, x10, x10 ∈ {x11 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x5) (Sing x5))|equip ... ...} leaving 2 subgoals.
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