Let x0 of type ι be given.
Let x1 of type ι be given.
Apply unknownprop_f82d0f217e1b2a36bc273d145ee21e9b9e753d654bb0c650cc08860c1b4bd1f0 with
x0,
x1,
equip (setprod 1 x0) x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x2 of type ι → ι be given.
Apply unknownprop_db24d9aa1dc52b3c0eaf7cf69655226164a8ab5afc5d72e14a32016133f537ca with
x0,
x1,
x2,
equip (setprod 1 x0) x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H3:
∀ x3 . In x3 x1 ⟶ ∃ x4 . and (In x4 x0) (x2 x4 = x3).
Apply unknownprop_6a8f953ba7c3bf327e583b76a91b24ddd499843a498fbfe2514e26f3800e68b3 with
x0,
x1,
x2,
equip (setprod 1 x0) x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4:
∀ x3 . In x3 x0 ⟶ In (x2 x3) x1.
Assume H5:
∀ x3 . In x3 x0 ⟶ ∀ x4 . In x4 x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Apply unknownprop_4b95783dcb3eee1943e1de5542f675166ef402c8fbdda80bdf0920b55d3fc6de with
setprod 1 x0,
x1,
λ x3 . x2 (proj1 x3).
Apply unknownprop_aa42ade5598d8612d2029318c4ed81646c550ecc6cdd9ab953ce4bf73f3dd562 with
setprod 1 x0,
x1,
λ x3 . x2 (proj1 x3) leaving 2 subgoals.
Apply unknownprop_57c8600e4bc6abecef2ae17962906fa2de1fc16f5d46ed100ff99cd5b67f5b1b with
setprod 1 x0,
x1,
λ x3 . x2 (proj1 x3) leaving 2 subgoals.
Let x3 of type ι be given.
Apply H4 with
proj1 x3.
Apply unknownprop_0ca88b304f8486344af25c1dccffdd0de0911b83400025c64b3ac2bfa45e498f with
1,
x0,
x3.
The subproof is completed by applying H6.
Let x3 of type ι be given.
Let x4 of type ι be given.
Apply unknownprop_25d0316470b9bdc33df1b5827718337aefe32f6ee5207178fa8d15f5c0f986af with
1,
x0,
x3,
λ x5 x6 . x5 = x4 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply unknownprop_25d0316470b9bdc33df1b5827718337aefe32f6ee5207178fa8d15f5c0f986af with
1,
x0,
x4,
λ x5 x6 . lam 2 (λ x7 . If_i (x7 = 0) (ap x3 0) (ap x3 1)) = x5 leaving 2 subgoals.
The subproof is completed by applying H7.
set y5 to be ...
set y6 to be
λ x6 x7 . lam 2 (λ x8 . If_i (x8 = 0) x7 (ap x4 1)) = lam 2 (λ x8 . If_i (x8 = 0) (ap y5 0) ...)Apply L9 with
λ x7 . y6 x7 y5 ⟶ y6 y5 x7 leaving 2 subgoals.
Assume H10: y6 y5 y5.
The subproof is completed by applying H10.
Apply unknownprop_3030666c70e3b0a40e79c3dea18074452b4f067cae53efb13b5a85e3938a9613 with
y5,
λ x7 x8 . x7 = ap y6 1,
λ x7 x8 . lam 2 (λ x9 . If_i (x9 = 0) (ap y6 0) x8) = lam 2 (λ x9 . If_i (x9 = 0) (ap y6 0) (ap y6 1)) leaving 2 subgoals.
Apply unknownprop_3030666c70e3b0a40e79c3dea18074452b4f067cae53efb13b5a85e3938a9613 with
y6,
λ x7 x8 . proj1 y5 = x7.
Apply H6 with
proj1 y5,
proj1 y6 leaving 3 subgoals.
Apply unknownprop_0ca88b304f8486344af25c1dccffdd0de0911b83400025c64b3ac2bfa45e498f with
1,
x2,
y5.
The subproof is completed by applying H7.
Apply unknownprop_0ca88b304f8486344af25c1dccffdd0de0911b83400025c64b3ac2bfa45e498f with
1,
x2,
y6.
The subproof is completed by applying H8.
The subproof is completed by applying L9.
Let x7 of type ι → ι → ο be given.
Assume H10:
x7 (lam 2 (λ x8 . If_i (x8 = 0) (ap y6 0) (ap y6 1))) (lam 2 (λ x8 . If_i (x8 = 0) (ap y6 0) (ap y6 1))).
The subproof is completed by applying H10.