Let x0 of type ι be given.

Assume H0: 8160a.. x0.

Apply H0 with λ x1 . x1 = 50941.. (f482f.. x1 4a7ef..) (e3162.. (f482f.. x1 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x1 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x1 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x1 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))).

Let x1 of type ι be given.

Let x2 of type ι → ι → ι be given.

Assume H1: ∀ x3 . prim1 x3 x1 ⟶ ∀ x4 . prim1 x4 x1 ⟶ prim1 (x2 x3 x4) x1.

Let x3 of type ι → ι → ο be given.

Let x4 of type ι be given.

Assume H2: prim1 x4 x1.

Let x5 of type ι be given.

Assume H3: prim1 x5 x1.

Apply unknownprop_4d3b1552f811436b8f29af8fa2db6a6f0ed851417fe4ea754fadac215fbbceef with x1, x2, x3, x4, x5, λ x6 x7 . 50941.. x1 x2 x3 x4 x5 = 50941.. x6 (e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))).

Apply unknownprop_ddb82a3c52c030568b927ce40224fa285ba790e62b2413b7fca117d1b06927bc with x1, x2, x3, x4, x5, λ x6 x7 . 50941.. x1 x2 x3 x4 x5 = 50941.. x1 (e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))).

Apply unknownprop_a2784db0fbde3c5924110ee44b88db7f7be64c189a40d459e45f4debf5b5aa45 with x1, x2, x3, x4, x5, λ x6 x7 . 50941.. x1 x2 x3 x4 x5 = 50941.. x1 (e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x6.

Apply unknownprop_f8827fd02e92f9891f514baa384373adf5f12cd2898ce08d25753ee7c26afa51 with x1, x2, e3162.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)), x3, 2b2e3.. (f482f.. (50941.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))), x4, x5 leaving 2 subgoals.

The subproof is completed by applying unknownprop_1577ea96652eddec718222b40e2ddf7792be06e6a335faa6da4659776d8b2b7f with x1, x2, x3, x4, x5.

Let x6 of type ι be given.

Assume H4: prim1 x6 x1.

Let x7 of type ι be given.

Assume H5: prim1 x7 x1.

Apply unknownprop_303a5dcd96ccd0795a0c1c2f0ababef222f2dd40fc6be62386d34762cfdb65c5 with x1, x2, x3, x4, x5, x6, x7, λ x8 x9 : ο . iff (x3 x6 x7) x8 leaving 3 subgoals.

The subproof is completed by applying H4.

The subproof is completed by applying H5.

The subproof is completed by applying iff_refl with x3 x6 x7.

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Proofgold Proof