Let x0 of type ι → (ι → ο) → ο be given.

Let x1 of type ι → (ι → ο) → ο be given.

Assume H0: ed32f.. x0 x1.

Let x2 of type ι be given.

Assume H1: ordinal x2.

Assume H2: PNo_lenbdd x2 x0.

Assume H3: PNo_lenbdd x2 x1.

Claim L4: ...

...

Apply L4 with ∃ x3 . and (∃ x4 : ι → ο . ced99.. x0 x1 x3 x4) (∀ x4 x5 : ι → ο . ced99.. x0 x1 x3 x4 ⟶ ced99.. x0 x1 x3 x5 ⟶ x4 = x5).

Let x3 of type ι be given.

Assume H5: (λ x4 . and (and (ordinal x4) (∃ x5 : ι → ο . 47618.. x0 x1 x4 x5)) (∀ x5 . prim1 x5 x4 ⟶ not (∃ x6 : ι → ο . 47618.. x0 x1 x5 x6))) x3.

Apply H5 with ∃ x4 . and (∃ x5 : ι → ο . ced99.. x0 x1 x4 x5) (∀ x5 x6 : ι → ο . ced99.. x0 x1 x4 x5 ⟶ ced99.. x0 x1 x4 x6 ⟶ x5 = x6).

Assume H6: and (ordinal x3) (∃ x4 : ι → ο . 47618.. x0 x1 x3 x4).

Apply H6 with (∀ x4 . prim1 x4 x3 ⟶ not (∃ x5 : ι → ο . 47618.. x0 x1 x4 x5)) ⟶ ∃ x4 . and (∃ x5 : ι → ο . ced99.. x0 x1 x4 x5) (∀ x5 x6 : ι → ο . ced99.. x0 x1 x4 x5 ⟶ ced99.. x0 x1 x4 x6 ⟶ x5 = x6).

Assume H7: ordinal x3.

Assume H8: ∃ x4 : ι → ο . 47618.. x0 x1 x3 x4.

Assume H9: ∀ x4 . prim1 x4 x3 ⟶ not (∃ x5 : ι → ο . 47618.. x0 x1 x4 x5).

Apply H8 with ∃ x4 . and (∃ x5 : ι → ο . ced99.. x0 x1 x4 x5) (∀ x5 x6 : ι → ο . ced99.. x0 x1 x4 x5 ⟶ ced99.. x0 x1 x4 x6 ⟶ x5 = x6).

Let x4 of type ι → ο be given.

Assume H10: 47618.. x0 x1 x3 x4.

Let x5 of type ο be given.

Assume H11: ∀ x6 . and (∃ x7 : ι → ο . ced99.. x0 x1 x6 x7) (∀ x7 x8 : ι → ο . ced99.. x0 x1 x6 x7 ⟶ ced99.. x0 x1 x6 x8 ⟶ x7 = x8) ⟶ x5.

Apply H11 with x3.

Apply andI with ∃ x6 : ι → ο . ced99.. x0 x1 x3 x6, ∀ x6 x7 : ι → ο . ced99.. x0 x1 x3 x6 ⟶ ced99.. x0 x1 x3 x7 ⟶ x6 = x7 leaving 2 subgoals.

Let x6 of type ο be given.

Assume H12: ∀ x7 : ι → ο . ced99.. x0 x1 x3 x7 ⟶ x6.

Apply H12 with λ x7 . and (prim1 x7 x3) (x4 x7).

Apply andI with 1a487.. x0 x1 x3 (λ x7 . and (prim1 x7 x3) (x4 x7)), ∀ x7 . nIn x7 ... ⟶ not (and (prim1 x7 x3) (x4 x7)) leaving 2 subgoals.

...

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Proofgold Proof