Let x0 of type ι → ((ι → ο) → ο) → (ι → ι → ι) → ι → 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 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) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4.
Apply unknownprop_0e532d62b1ffeb2e44a653293314211bf6ef2f87c27f36efb1563be78fd78d6d with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (decode_c (f482f.. (e8b89.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (e8b89.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (e8b89.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x0 x1 x2 x3 x4.
Apply unknownprop_461a5994de8b4df809b00ca38ec0a04acad276e10d8366f13b7e301477945ae4 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x1 (decode_c (f482f.. (e8b89.. x1 x2 x3 x4) (4ae4a.. 4a7ef..))) (e3162.. (f482f.. (e8b89.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..)))) x5 = x0 x1 x2 x3 x4.
Apply H0 with
decode_c (f482f.. (e8b89.. x1 x2 x3 x4) (4ae4a.. 4a7ef..)),
e3162.. (f482f.. (e8b89.. x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) leaving 2 subgoals.
Let x5 of type ι → ο be given.
Assume H1:
∀ x6 . x5 x6 ⟶ prim1 x6 x1.
Apply unknownprop_87bcd25fc3366287afa74e30e32811ff9f473a08c1b1dad134df5a6438372c7f 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_11a43fb112690babe480ed081c340e61eff9ede8136723ff028f6405ad595668 with x1, x2, x3, x4.