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Proofgold Proof

pf
Let x0 of type ι(ιιιι) → ιιι be given.
Assume H0: ∀ x1 . ∀ x2 x3 : ι → ι → ι → ι . (∀ x4 . x4x1x2 x4 = x3 x4)x0 x1 x2 = x0 x1 x3.
Let x1 of type ι be given.
Let x2 of type ι(ιιι) → ο be given.
Assume H1: ∀ x3 . ∀ x4 : ι → ι → ι → ι . (∀ x5 . x5x3x2 x5 (x4 x5))x2 x3 (x0 x3 x4).
Apply H1 with x1, In_rec_iii x0.
Let x3 of type ι be given.
Assume H2: x3x1.
Apply In_rec_G_iii_In_rec_iii with x0, x3, x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.