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 ι be given.
Let x15 of type ι be given.
Let x16 of type ι be given.
Let x17 of type ι be given.
Let x18 of type ι be given.
Assume H5:
Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.
Assume H6:
not (x6 x18 (x1 (x3 x4 x14) (x9 x15 x16 x14)) x17 = x4).
Apply unknownprop_f7b5ec75bad85de491ae811af7281b8523890ceb7b0a21e077fce8976ff58a01 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13 leaving 2 subgoals.
The subproof is completed by applying H5.
Let x19 of type ο be given.
Assume H7:
∀ x20 . and (In x20 x0) (∃ x21 . and (In x21 x0) (∃ x22 . and (In x22 x0) (∃ x23 . and (In x23 x0) (∃ x24 . and (In x24 x0) (not (x6 x24 (x1 (x3 x4 x20) (x9 x21 x22 x20)) x23 = x4)))))) ⟶ x19.
Apply H7 with
x14.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x14 x0,
∃ x20 . and (In x20 x0) (∃ x21 . and (In x21 x0) (∃ x22 . and (In x22 x0) (∃ x23 . and (In x23 x0) (not (x6 x23 (x1 (x3 x4 x14) (x9 x20 x21 x14)) x22 = x4))))) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x20 of type ο be given.
Assume H8:
∀ x21 . and (In x21 x0) (∃ x22 . and (In x22 x0) (∃ x23 . and (In x23 x0) (∃ x24 . and (In x24 x0) (not (x6 x24 (x1 (x3 x4 x14) (x9 x21 x22 x14)) x23 = x4))))) ⟶ x20.
Apply H8 with
x15.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x15 x0,
∃ x21 . and (In x21 x0) (∃ x22 . and (In x22 x0) (∃ x23 . and (In x23 x0) (not (x6 x23 (x1 (x3 x4 x14) (x9 x15 x21 x14)) x22 = x4)))) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x21 of type ο be given.
Assume H9:
∀ x22 . and (In x22 x0) (∃ x23 . and (In x23 x0) (∃ x24 . and (In x24 x0) (not (x6 x24 (x1 (x3 x4 x14) (x9 x15 x22 x14)) x23 = x4)))) ⟶ x21.
Apply H9 with
x16.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x16 x0,
∃ x22 . and (In x22 x0) (∃ x23 . ...) leaving 2 subgoals.