Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ο be given.
Apply H0 with
λ x4 . x4 = 1670d.. x0 x1 x2 x3 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 leaving 2 subgoals.
Let x4 of type ι be given.
Let x5 of type ι → ι → ι be given.
Let x6 of type ι → ι be given.
Let x7 of type ι → ο be given.
Apply unknownprop_abde0c95efb9c5a36a3b1b60fce4f97396bb4d91574381fb7896801234f8a172 with
x4,
x0,
x5,
x1,
x6,
x2,
x7,
x3,
∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H4:
and (and (x4 = x0) (∀ x8 . prim1 x8 x4 ⟶ ∀ x9 . prim1 x9 x4 ⟶ x5 x8 x9 = x1 x8 x9)) (∀ x8 . prim1 x8 x4 ⟶ x6 x8 = x2 x8).
Apply H4 with
(∀ x8 . prim1 x8 x4 ⟶ x7 x8 = x3 x8) ⟶ ∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0.
Assume H5:
and (x4 = x0) (∀ x8 . prim1 x8 x4 ⟶ ∀ x9 . prim1 x9 x4 ⟶ x5 x8 x9 = x1 x8 x9).
Apply H5 with
(∀ x8 . prim1 x8 x4 ⟶ x6 x8 = x2 x8) ⟶ (∀ x8 . prim1 x8 x4 ⟶ x7 x8 = x3 x8) ⟶ ∀ x8 . prim1 x8 x0 ⟶ prim1 (x2 x8) x0.
Assume H6: x4 = x0.
Assume H7:
∀ x8 . prim1 x8 x4 ⟶ ∀ x9 . prim1 x9 x4 ⟶ x5 x8 x9 = x1 x8 x9.
Assume H8:
∀ x8 . prim1 x8 x4 ⟶ x6 x8 = x2 x8.
Assume H9:
∀ x8 . prim1 x8 x4 ⟶ x7 x8 = x3 x8.
Apply H6 with
λ x8 x9 . ∀ x10 . prim1 x10 x8 ⟶ prim1 (x2 x10) x8.
Let x8 of type ι be given.
Apply H8 with
x8,
λ x9 x10 . prim1 x9 x4 leaving 2 subgoals.
The subproof is completed by applying H10.
Apply H2 with
x8.
The subproof is completed by applying H10.
Let x4 of type ι → ι → ο be given.
The subproof is completed by applying H1.