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.
Let x14 of type ο be given.
Assume H1:
∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ ∀ x18 . In x18 x0 ⟶ Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ not (x5 (x1 (x2 (x8 x16 x17 x15) x4) x15) x18 = x4) ⟶ x14.
Apply unknownprop_49eb140bf3d095f83f1559108a1206c9ee6f7677038bb9d3263a8d231bb3c9c3 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2:
Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
Assume H3:
∃ x15 . and (In x15 x0) (∃ x16 . and (In x16 x0) (∃ x17 . and (In x17 x0) (∃ x18 . and (In x18 x0) (not (x5 (x1 (x2 (x8 x16 x17 x15) x4) x15) x18 = x4))))).
Apply H3 with
x14.
Let x15 of type ι be given.
Apply unknownprop_670c1bb10dc1952c71f5fdb407208a9646b2ed4c350ab9dc752cc19ec9535b95 with
In x15 x0,
∃ x16 . and (In x16 x0) (∃ x17 . and (In x17 x0) (∃ x18 . and (In x18 x0) (not (x5 (x1 (x2 (x8 x16 x17 x15) x4) x15) x18 = x4)))),
x14.
Assume H5:
∃ x16 . and (In x16 x0) (∃ x17 . and (In x17 x0) (∃ x18 . and (In x18 x0) (not (x5 (x1 (x2 (x8 x16 x17 x15) x4) x15) x18 = x4)))).
Apply H5 with
x14.
Let x16 of type ι be given.
Apply unknownprop_670c1bb10dc1952c71f5fdb407208a9646b2ed4c350ab9dc752cc19ec9535b95 with
In x16 x0,
∃ x17 . and (In x17 x0) (∃ x18 . and (In x18 x0) (not (x5 (x1 (x2 (x8 x16 x17 x15) x4) x15) x18 = x4))),
x14.
Assume H7:
∃ x17 . and (In x17 x0) (∃ x18 . and (In x18 x0) (not (x5 (x1 (x2 (x8 x16 x17 x15) x4) x15) x18 = x4))).
Apply H7 with
x14.
Let x17 of type ι be given.
Apply unknownprop_670c1bb10dc1952c71f5fdb407208a9646b2ed4c350ab9dc752cc19ec9535b95 with
In x17 x0,
∃ x18 . and (In x18 ...) ...,
....