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 → ι → ο be given.
Let x4 of type ι be given.
Apply H0 with
λ x5 . x5 = 04077.. 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 → ι → ο be given.
Let x9 of type ι be given.
Apply unknownprop_8d3c02258783c53c03a47d97ca3196fe290f0aea6e3393666b2e7cc477a3fb91 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 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x8 x10 x11 = x3 x10 x11).
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 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x8 x10 x11 = x3 x10 x11) ⟶ 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 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x8 x10 x11 = x3 x10 x11) ⟶ 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 . ... ⟶ ∀ x11 . ... ⟶ x8 x10 x11 = ....