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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Assume H0: ∀ x2 . x1 x2 ⟶ ∀ x3 . x3 ∈ x2 ⟶ nIn x0 x3.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι → ι → ι be given.
Assume H1: ∀ x6 . x1 x6 ⟶ x1 (x2 x6).
Assume H2: ∀ x6 . x1 x6 ⟶ x1 (x3 x6).
Assume H3: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x1 (x4 x6 x7).
Assume H4: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x1 (x5 x6 x7).
Assume H5: ∀ x6 . x1 x6 ⟶ x3 (x2 x6) = x2 (x3 x6).
Assume H6: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x2 (x4 x6 x7) = x4 (x2 x6) (x2 x7).
Assume H7: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x5 x6 (x2 x7) = x2 (x5 x6 x7).
Assume H8: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x5 (x2 x6) x7 = x2 (x5 x6 x7).
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H9: CD_carr x0 x1 x6.
Assume H10: CD_carr x0 x1 x7.
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Apply CD_proj0_2 with x0, x1, x2 (CD_proj0 x0 x1 x7), x2 (CD_proj1 x0 x1 x7), λ x8 x9 . pair_tag x0 (x4 (x5 (CD_proj0 x0 x1 x6) x9) (x2 (x5 (x3 (CD_proj1 x0 x1 (pair_tag x0 (x2 (CD_proj0 x0 x1 x7)) (x2 (CD_proj1 x0 x1 x7))))) (CD_proj1 x0 x1 x6)))) (x4 (x5 (CD_proj1 x0 x1 (pair_tag x0 (x2 (CD_proj0 x0 x1 x7)) (x2 (CD_proj1 x0 x1 x7)))) (CD_proj0 x0 x1 x6)) (x5 (CD_proj1 x0 x1 x6) (x3 x9))) = pair_tag x0 (x2 (CD_proj0 x0 x1 (pair_tag x0 (x4 (x5 (CD_proj0 x0 x1 x6) (CD_proj0 x0 x1 x7)) (x2 (x5 (x3 (CD_proj1 x0 x1 x7)) (CD_proj1 x0 x1 x6)))) (x4 (x5 (CD_proj1 x0 x1 x7) (CD_proj0 x0 x1 x6)) (x5 (CD_proj1 x0 x1 x6) (x3 (CD_proj0 x0 x1 x7))))))) (x2 (CD_proj1 x0 x1 (pair_tag x0 (x4 (x5 (CD_proj0 x0 x1 x6) (CD_proj0 x0 x1 x7)) (x2 (x5 (x3 (CD_proj1 x0 x1 x7)) ...))) ...))) leaving 4 subgoals.
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