Let x0 of type ι be given.
Let x1 of type (ι → ο) → ο be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ο be given.
Let x4 of type ι be given.
Apply H0 with
λ x5 . x5 = bebf6.. x0 x1 x2 x3 x4 ⟶ prim1 x4 x0 leaving 2 subgoals.
Let x5 of type ι be given.
Let x6 of type (ι → ο) → ο be given.
Let x7 of type ι → ι → ι be given.
Let x8 of type ι → ο be given.
Let x9 of type ι be given.
Apply unknownprop_1c9afda7d2e7e676714a733a3d4b91747d862269cb9abc70ac5847feacc6f2d6 with
x5,
x0,
x6,
x1,
x7,
x2,
x8,
x3,
x9,
x4,
prim1 x4 x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H4:
and (and (and (x5 = x0) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x5) ⟶ x6 x10 = x1 x10)) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11)) (∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10).
Apply H4 with
x9 = x4 ⟶ prim1 x4 x0.
Assume H5:
and (and (x5 = x0) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x5) ⟶ x6 x10 = x1 x10)) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11).
Apply H5 with
(∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10) ⟶ x9 = x4 ⟶ prim1 x4 x0.
Assume H6:
and (x5 = x0) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x5) ⟶ x6 x10 = x1 x10).
Apply H6 with
(∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11) ⟶ (∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10) ⟶ x9 = x4 ⟶ prim1 x4 x0.
Assume H7: x5 = x0.
Assume H8:
∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x5) ⟶ x6 x10 = x1 x10.
Assume H9:
∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x7 x10 x11 = x2 x10 x11.
Assume H10:
∀ x10 . prim1 x10 x5 ⟶ x8 x10 = x3 x10.
Assume H11: x9 = x4.
Apply H7 with
λ x10 x11 . prim1 x4 x10.
Apply H11 with
λ x10 x11 . prim1 x10 x5.
The subproof is completed by applying H2.