Let x0 of type ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι → ο be given.
Assume H0:
∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4.
Apply unknownprop_ba81b186b20404f2b184823fc867586e7d7244eedb795612a8200f2a550b9d96 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (decode_c (f482f.. (d7d7e.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (d7d7e.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (d7d7e.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4.
Apply H0 with
decode_c (f482f.. (d7d7e.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
e3162.. (f482f.. (d7d7e.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (d7d7e.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
Let x5 of type ι → ο be given.
Assume H1:
∀ x6 . x5 x6 ⟶ prim1 x6 x1.
Apply unknownprop_7b7686cbecedcd60f8f5fdb253b765355b51e9838870215bb6d8f625abc19e28 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x2 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x5.
The subproof is completed by applying unknownprop_da5395658eb11644493773387119e852dc723b35ba38d26d0dcd5c668a45e5db with x1, x2, x3, x4.
Let x5 of type ι be given.
Apply unknownprop_b00f9274fcdc66a19bf75646be0063284f767971ab29262c420369ce1337f459 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x4 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x5.