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 → ι → ο be given.
Let x4 of type ι → ι → ο be given.
Apply H0 with
λ x5 . x5 = d0dcf.. x0 x1 x2 x3 x4 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x6) x0 leaving 2 subgoals.
Let x5 of type ι be given.
Let x6 of type ι → ι → ι be given.
Let x7 of type ι → ι be given.
Let x8 of type ι → ι → ο be given.
Let x9 of type ι → ι → ο be given.
Apply unknownprop_cdd5b95163c0a89a0ba14d62cf34277886275d7a6d60a8db89301c0c832d8e9f with
x5,
x0,
x6,
x1,
x7,
x2,
x8,
x3,
x9,
x4,
∀ x10 . prim1 x10 x0 ⟶ prim1 (x2 x10) x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H4:
and (and (and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x6 x10 x11 = x1 x10 x11)) (∀ x10 . prim1 x10 x5 ⟶ x7 x10 = x2 x10)) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x8 x10 x11 = x3 x10 x11).
Apply H4 with
(∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x9 x10 x11 = x4 x10 x11) ⟶ ∀ x10 . prim1 x10 x0 ⟶ prim1 (x2 x10) x0.
Assume H5:
and (and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x6 x10 x11 = x1 x10 x11)) (∀ x10 . prim1 x10 x5 ⟶ x7 x10 = x2 x10).
Apply H5 with
(∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x8 x10 x11 = x3 x10 x11) ⟶ (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x9 x10 x11 = x4 x10 x11) ⟶ ∀ x10 . prim1 x10 x0 ⟶ prim1 (x2 x10) x0.
Assume H6:
and (x5 = x0) (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x6 x10 x11 = x1 x10 x11).
Apply H6 with
... ⟶ ... ⟶ (∀ x10 . prim1 x10 x5 ⟶ ∀ x11 . prim1 x11 x5 ⟶ x9 x10 x11 = x4 x10 x11) ⟶ ∀ x10 . prim1 x10 x0 ⟶ prim1 (x2 x10) x0.