Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι be given.
Let x5 of type ι → ι → ι be given.
Let x6 of type ι → ι → ι → ι be given.
Let x7 of type ι → ι → ι be given.
Let x8 of type ι → ι → ι → ι be given.
Let x9 of type ι → ι → ι → ι be given.
Let x10 of type ι → ι → ι be given.
Let x11 of type ι → ι → ι be given.
Let x12 of type ι → ι → ι be given.
Let x13 of type ι → ι → ι be given.
Assume H0:
Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
Let x14 of type ο be given.
Assume H2:
(∀ x15 . In x15 x0 ⟶ x2 x4 x15 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x2 x15 x15 = x4) ⟶ (∀ x15 . In x15 x0 ⟶ x3 x15 x4 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x3 x15 x15 = x4) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x8 x4 x15 x16 = x16) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x8 x15 x4 x16 = x16) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x9 x4 x15 x16 = x16) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x9 x15 x4 x16 = x16) ⟶ (∀ x15 . In x15 x0 ⟶ x7 x4 x15 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x10 x4 x15 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x11 x4 x15 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x12 x4 x15 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x13 x4 x15 = x15) ⟶ x14.
Apply unknownprop_aa728e7fe7e33e3e3587a7ee1909d24907e5e9713406d48a03419b0711541ca1 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H3:
∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x1 x15 x16) x0.
Assume H4:
∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x2 x15 x16) x0.
Assume H5:
∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x3 x15 x16) x0.
Assume H6:
∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x7 x15 x16) x0.
Assume H7:
∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ In (x8 x15 x16 x17) x0.
Assume H8:
∀ x15 . ... ⟶ ∀ x16 . ... ⟶ ∀ x17 . In x17 ... ⟶ In (x9 x15 x16 x17) x0.