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Proofgold Proof

pf
Let x0 of type ι(ι((ιο) → ο) → ο) → ((ιο) → ο) → ο be given.
Assume H0: ∀ x1 . ∀ x2 x3 : ι → ((ι → ο) → ο) → ο . (∀ x4 . prim1 x4 x1x2 x4 = x3 x4)x0 x1 x2 = x0 x1 x3.
Apply In_ind with λ x1 . 1d01c.. x0 x1 (In_rec_Vo3 x0 x1).
Let x1 of type ι be given.
Assume H1: ∀ x2 . prim1 x2 x11d01c.. x0 x2 (In_rec_Vo3 x0 x2).
Apply Descr_Vo3_prop with 1d01c.. x0 x1 leaving 2 subgoals.
Let x2 of type ο be given.
Assume H2: ∀ x3 : ((ι → ο) → ο) → ο . 1d01c.. x0 x1 x3x2.
Apply H2 with x0 x1 (In_rec_Vo3 x0).
Apply unknownprop_19e14cfefa9c62802c3472f7b613074ec9e1720289fb25abc9d15e165cea68de with x0, x1, In_rec_Vo3 x0.
The subproof is completed by applying H1.
Apply unknownprop_e3fd7e34e0da9eea44164f3141ff6fc232704ab917a0a07eb8451b62e6e7b7c2 with x0, x1.
The subproof is completed by applying H0.