Let x0 of type ι be given.
Let x1 of type ι be given.
Apply unknownprop_4b95783dcb3eee1943e1de5542f675166ef402c8fbdda80bdf0920b55d3fc6de with
setexp x0 (ordsucc x1),
setprod (setexp x0 x1) x0,
λ x2 . lam 2 (λ x3 . If_i (x3 = 0) (lam x1 (λ x4 . ap x2 x4)) (ap x2 x1)).
Apply unknownprop_aa42ade5598d8612d2029318c4ed81646c550ecc6cdd9ab953ce4bf73f3dd562 with
setexp x0 (ordsucc x1),
setprod (setexp x0 x1) x0,
λ x2 . lam 2 (λ x3 . If_i (x3 = 0) (lam x1 (λ x4 . ap x2 x4)) (ap x2 x1)) leaving 2 subgoals.
Apply unknownprop_57c8600e4bc6abecef2ae17962906fa2de1fc16f5d46ed100ff99cd5b67f5b1b with
setexp x0 (ordsucc x1),
setprod (setexp x0 x1) x0,
λ x2 . lam 2 (λ x3 . If_i (x3 = 0) (lam x1 (λ x4 . ap x2 x4)) (ap x2 x1)) leaving 2 subgoals.
The subproof is completed by applying L0.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H3:
(λ x4 . lam 2 (λ x5 . If_i (x5 = 0) (lam x1 (λ x6 . ap x4 x6)) (ap x4 x1))) x2 = (λ x4 . lam 2 (λ x5 . If_i (x5 = 0) (lam x1 (λ x6 . ap x4 x6)) (ap x4 x1))) x3.
Apply unknownprop_23208921203993e7c79234f69a10e3d42c3011a560c83fb48a9d1a8f3b50675c with
ordsucc x1,
x0,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x4 of type ι be given.
Apply unknownprop_84fe37a922385756a4e0826a593defb788cadbe4bdc9a7fe6b519ea49f509df5 with
x1,
x4,
ap x2 x4 = ap x3 x4 leaving 3 subgoals.
The subproof is completed by applying H4.
Apply unknownprop_c5e2164052a280ad5b04f622e53815f0267ee33361e4345305e43303abef2c1b with
x1,
λ x5 . ap x2 x5,
x4,
λ x5 x6 . x5 = ap x3 x4 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply unknownprop_c5e2164052a280ad5b04f622e53815f0267ee33361e4345305e43303abef2c1b with
x1,
λ x5 . ap x3 x5,
x4,
λ x5 x6 . ap (lam x1 (ap x2)) x4 = x5 leaving 2 subgoals.
The subproof is completed by applying H5.
Claim L6:
lam x1 (λ x5 . ap x2 x5) = lam x1 (λ x5 . ap x3 x5)Apply unknownprop_ab640183089126bd192ea777bf3d0693f7fc019527e86171644bd6e54256c0e4 with
lam x1 (λ x5 . ap x3 x5),
ap x3 x1,
λ x5 x6 . lam x1 (λ x7 . ap x2 x7) = x5.
Apply unknownprop_ab640183089126bd192ea777bf3d0693f7fc019527e86171644bd6e54256c0e4 with
lam x1 (λ x5 . ap x2 x5),
ap x2 x1,
λ x5 x6 . x5 = ap ((λ x7 . lam 2 (λ x8 . If_i (x8 = 0) (lam ... ...) ...)) ...) 0.
Apply L6 with
λ x5 x6 . ap x6 x4 = ap (lam x1 (λ x7 . ap x3 x7)) x4.
Let x5 of type ι → ι → ο be given.
Assume H7:
x5 (ap (lam x1 (λ x6 . ap x3 x6)) x4) (ap (lam x1 (λ x6 . ap x3 x6)) x4).
The subproof is completed by applying H7.