Let x0 of type ι → ((ι → ο) → ο) → (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 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 . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4.
Apply unknownprop_8ea88471b9883b8abffe33a1bf4f2c8ee44940148303bba54f55e7f53dcad418 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (decode_c (f482f.. (c7ccc.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (c7ccc.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (c7ccc.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) = x0 x1 x2 x3 x4.
Apply H0 with
decode_c (f482f.. (c7ccc.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
f482f.. (f482f.. (c7ccc.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))),
decode_p (f482f.. (c7ccc.. 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_98b446a891a5eaf8b34ebae3d85a1a49a0be454d25581266db000e1a95a5a7fd 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_29ae15428d4a6af9f40e9d1bd723bdbf6b510330107b33f5917ed46c37fbc7a8 with x1, x2, x3, x4.
Let x5 of type ι be given.
Apply unknownprop_3b4756d0bcc65fbb21c27a602fad093f481516dd6337133dc11f6eaeb19419b0 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.