Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Assume H1:
∀ x3 . In x3 x1 ⟶ nIn x3 x0 ⟶ In 0 (x2 x3).
Apply unknownprop_c3fe42b21df0810041479a97b374de73f7754e07c8af1c88386a1e7dc0aad10f with
Pi x0 (λ x3 . x2 x3),
Pi x1 (λ x3 . x2 x3).
Let x3 of type ι be given.
Assume H2:
In x3 (Pi x0 (λ x4 . x2 x4)).
Apply unknownprop_c20579f7ec03c9b411c1afcdcbd6eb7f887b4dea35b13dd2fe5a71172b6554fe with
x0,
x2,
x3,
In x3 (Pi x1 (λ x4 . x2 x4)) leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4:
∀ x4 . In x4 x0 ⟶ In (ap x3 x4) (x2 x4).
Apply unknownprop_1a1eed9c2e0652a509eabe7b8f07e31768cab0357ad1d97cb464202e3d371a17 with
x1,
x2,
x3 leaving 2 subgoals.
Let x4 of type ι be given.
Apply andE with
setsum_p x4,
In (ap x4 0) x0,
and (setsum_p x4) (In (ap x4 0) x1) leaving 2 subgoals.
Apply H3 with
x4.
The subproof is completed by applying H5.
Assume H7:
In (ap x4 0) x0.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
setsum_p x4,
In (ap x4 0) x1 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply unknownprop_cc8f63ddfbec05087d89028647ba2c7b89da93a15671b61ba228d6841bbab5e9 with
x0,
x1,
ap x4 0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H7.
Let x4 of type ι be given.
Apply unknownprop_80056f9db85f84f8ce0644e1cdb62f5e66ec62c041f28dd7a07b3c46de1ea696 with
x4,
x0,
In (ap x3 x4) (x2 x4) leaving 2 subgoals.
Apply H4 with
x4.
The subproof is completed by applying H6.
Apply unknownprop_1d401bfd2bdb443cb4ed66a0412f4299139e285cca5e460f0f1c03467ae8078c with
x0,
x2,
x3,
λ x5 x6 . ap x5 x4 = 0 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_65bac37c150594021865ca4bcc52ede90a62ee904e2b827d4b233886c39d597e with
x0,
ap x3,
x4.
The subproof is completed by applying H6.
Apply L7 with
λ x5 x6 . In x6 (x2 x4).
Apply H1 with
x4 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.