Let x0 of type ι be given.
Let x1 of type (ι → ο) → ο be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι be given.
Apply H0 with
λ x4 . x4 = d8d01.. x0 x1 x2 x3 ⟶ prim1 x3 x0 leaving 2 subgoals.
Let x4 of type ι be given.
Let x5 of type (ι → ο) → ο be given.
Let x6 of type ι → ι → ο be given.
Let x7 of type ι be given.
Apply unknownprop_269415c66983514a2ee9992d7034521360ae834e287ae12f01a70f22c5611879 with
x4,
x0,
x5,
x1,
x6,
x2,
x7,
x3,
prim1 x3 x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3:
and (and (x4 = x0) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x4) ⟶ x5 x8 = x1 x8)) (∀ x8 . prim1 x8 x4 ⟶ ∀ x9 . prim1 x9 x4 ⟶ x6 x8 x9 = x2 x8 x9).
Apply H3 with
x7 = x3 ⟶ prim1 x3 x0.
Assume H4:
and (x4 = x0) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x4) ⟶ x5 x8 = x1 x8).
Apply H4 with
(∀ x8 . prim1 x8 x4 ⟶ ∀ x9 . prim1 x9 x4 ⟶ x6 x8 x9 = x2 x8 x9) ⟶ x7 = x3 ⟶ prim1 x3 x0.
Assume H5: x4 = x0.
Assume H6:
∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x4) ⟶ x5 x8 = x1 x8.
Assume H7:
∀ x8 . prim1 x8 x4 ⟶ ∀ x9 . prim1 x9 x4 ⟶ x6 x8 x9 = x2 x8 x9.
Assume H8: x7 = x3.
Apply H5 with
λ x8 x9 . prim1 x3 x8.
Apply H8 with
λ x8 x9 . prim1 x8 x4.
The subproof is completed by applying H1.
Let x4 of type ι → ι → ο be given.
The subproof is completed by applying H1.