Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply unknownprop_c5e2164052a280ad5b04f622e53815f0267ee33361e4345305e43303abef2c1b with
8,
λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) x1 (If_i (x8 = 2) x2 (If_i (x8 = 3) x3 (If_i (x8 = 4) x4 (If_i (x8 = 5) x5 (If_i (x8 = 6) x6 x7)))))),
3,
λ x8 x9 . x9 = x3 leaving 2 subgoals.
The subproof is completed by applying unknownprop_de650337eacddcedf94589db6d8c72ef3436c23cb10d098589fa2341bbc896dd.
Apply unknownprop_5a150bd86f4285de5d98c60b17d4452a655b4d88de0a02247259cdad6e6d992c with
3 = 0,
x0,
If_i (3 = 1) x1 (If_i (3 = 2) x2 (If_i (3 = 3) x3 (If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 x7))))),
λ x8 x9 . x9 = x3 leaving 2 subgoals.
The subproof is completed by applying unknownprop_e2fde4211108c28607e0761977b11b51c8c2832e6808f37d930cc361e0ac54cb.
Apply unknownprop_5a150bd86f4285de5d98c60b17d4452a655b4d88de0a02247259cdad6e6d992c with
3 = 1,
x1,
If_i (3 = 2) x2 (If_i (3 = 3) x3 (If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 x7)))),
λ x8 x9 . x9 = x3 leaving 2 subgoals.
The subproof is completed by applying unknownprop_43d49dcfa39552687d2ef1ea75fae8fe3937e3bbe00ca24b443502b5bed2c5b2.
Apply unknownprop_5a150bd86f4285de5d98c60b17d4452a655b4d88de0a02247259cdad6e6d992c with
3 = 2,
x2,
If_i (3 = 3) x3 (If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 x7))),
λ x8 x9 . x9 = x3 leaving 2 subgoals.
The subproof is completed by applying unknownprop_7aee809c6a95525e8e26d7e3157f4f94e1efac99d2380796f403e07122a12ae4.
Apply unknownprop_6f44febdf8a865ee94133af873e3c2941a931de6ac80968301360290e02ca608 with
3 = 3,
x3,
If_i (3 = 4) x4 (If_i (3 = 5) x5 (If_i (3 = 6) x6 x7)).
Let x8 of type ι → ι → ο be given.
Assume H0: x8 3 3.
The subproof is completed by applying H0.