Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: 80242.. x0.

Assume H1: 80242.. x1.

set y2 to be bc82c.. (bc82c.. x0 x1) (f4dc0.. x1)

set y3 to be x1

Claim L2: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2

Let x4 of type ι → ο be given.

Assume H2: x4 y2.

set y5 to be λ x5 . x4

Apply unknownprop_d8f468fc749efab866c779febbe4cd601b5e2eeaa90e3f207f17de20f4ab68ab with y2, y3, f4dc0.. y3, λ x6 x7 . y5 x7 x6 leaving 4 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Apply unknownprop_8fbd0c2b48e78882e15936a0ad5f4a3c5af2cb37d9320d8cb210a91462e202ca with y3.

The subproof is completed by applying H1.

set y6 to be bc82c.. y3 (bc82c.. x4 (f4dc0.. x4))

set y7 to be bc82c.. x4 4a7ef..

Claim L3: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6

Let x8 of type ι → ο be given.

Assume H3: x8 (bc82c.. y5 4a7ef..).

set y9 to be λ x9 . x8

Apply unknownprop_5cbe8643f1e45151334fcd7208ed4c5dc2b120486b5022cdc81170c910647c5d with y6, λ x10 x11 . y9 (bc82c.. y5 x10) (bc82c.. y5 x11) leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H3.

set y8 to be λ x8 . y7

Apply L3 with λ x9 . y8 x9 y7 ⟶ y8 y7 x9 leaving 2 subgoals.

Assume H4: y8 y7 y7.

The subproof is completed by applying H4.

Apply unknownprop_3201a7f403ac32d9752260880c29e9f9084a9b6a0cf9218208744c4c82940cb1 with y6, λ x9 . y8 leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying L3.

Let x4 of type ι → ι → ο be given.

Apply L2 with λ x5 . x4 x5 y3 ⟶ x4 y3 x5.

Assume H3: x4 y3 y3.

The subproof is completed by applying H3.

■

Proofgold Proof