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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H0: 02b90.. x0 x1.
Assume H1: 02b90.. x2 x3.
Claim L2: ...
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Claim L3: ...
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Claim L4: ...
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Apply H0 with 02b90.. (0ac37.. (94f9e.. x0 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x2 (λ x4 . bc82c.. (02a50.. x0 x1) x4))) (0ac37.. (94f9e.. x1 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x3 (λ x4 . bc82c.. (02a50.. x0 x1) x4))).
Assume H5: and (∀ x4 . prim1 x4 x0 ⟶ 80242.. x4) (∀ x4 . prim1 x4 x1 ⟶ 80242.. x4).
Apply H5 with (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x1 ⟶ 099f3.. x4 x5) ⟶ 02b90.. (0ac37.. (94f9e.. x0 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x2 (λ x4 . bc82c.. (02a50.. x0 x1) x4))) (0ac37.. (94f9e.. x1 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x3 (λ x4 . bc82c.. (02a50.. x0 x1) x4))).
Assume H6: ∀ x4 . prim1 x4 x0 ⟶ 80242.. x4.
Assume H7: ∀ x4 . prim1 x4 x1 ⟶ 80242.. x4.
Assume H8: ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x1 ⟶ 099f3.. x4 x5.
Apply H1 with 02b90.. (0ac37.. (94f9e.. x0 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x2 (λ x4 . bc82c.. (02a50.. x0 x1) x4))) (0ac37.. (94f9e.. x1 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x3 (λ x4 . bc82c.. (02a50.. x0 x1) x4))).
Assume H9: and (∀ x4 . prim1 x4 x2 ⟶ 80242.. x4) (∀ x4 . prim1 x4 x3 ⟶ 80242.. x4).
Apply H9 with (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x3 ⟶ 099f3.. x4 x5) ⟶ 02b90.. (0ac37.. (94f9e.. x0 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x2 (λ x4 . bc82c.. (02a50.. x0 x1) x4))) (0ac37.. (94f9e.. x1 (λ x4 . bc82c.. x4 (02a50.. x2 x3))) (94f9e.. x3 (λ x4 . bc82c.. (02a50.. x0 x1) x4))).
Assume H10: ∀ x4 . prim1 x4 x2 ⟶ 80242.. x4.
Assume H11: ∀ x4 . prim1 x4 x3 ⟶ 80242.. x4.
Assume H12: ∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x3 ⟶ 099f3.. x4 x5.
Apply and3I with ∀ x4 . ... ⟶ 80242.. x4, ..., ... leaving 3 subgoals.
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