Let x0 of type ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → ι → ι be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ο be given.
Let x5 of type ι be given.
Assume H0:
∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5.
Apply unknownprop_5091b23ace185de1439c9993ecf5d34c3889f1adbf8fc4314bc1ac1fed4972cf with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (decode_c (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4 x5.
Apply unknownprop_3ab7806cf0ff3d8bee638a9c7ebd5be410ba51ca429236345648ab7856c58a31 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (decode_c (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
decode_c (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)),
f482f.. (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) leaving 3 subgoals.
Let x6 of type ι → ο be given.
Assume H1:
∀ x7 . x6 x7 ⟶ prim1 x7 x1.
Apply unknownprop_a13b376ead7440cfb8068d51b0fc26b8bee9b2fd7f0c4983ba28169597604378 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x2 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x6.
The subproof is completed by applying unknownprop_08751a2182e392d896e3e3f4a36e5bf5e3e32f0eec6892705c231ffefc7cc6f0 with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Apply unknownprop_04e910d1de9e4552e9264adbcc682f73b876009ed78e0a9d5f152658b15ea3ba with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x4 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x6.