Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0:
∀ x3 . In x3 x0 ⟶ In (x2 x3) (x1 x3).
Apply unknownprop_1a1eed9c2e0652a509eabe7b8f07e31768cab0357ad1d97cb464202e3d371a17 with
x0,
λ x3 . x1 x3,
lam x0 (λ x3 . x2 x3) leaving 2 subgoals.
Let x3 of type ι be given.
Assume H1:
In x3 (lam x0 (λ x4 . x2 x4)).
Claim L2:
∃ x4 . and (In x4 x0) (∃ x5 . and (In x5 (x2 x4)) (x3 = setsum x4 x5))Apply unknownprop_f25818182af6b093121a8b5d43847162c8ea91396e524cca02557613a430a57a with
x0,
x2,
x3.
The subproof is completed by applying H1.
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x4 . In x4 x0,
λ x4 . ∃ x5 . and (In x5 (x2 x4)) (x3 = setsum x4 x5),
and (setsum_p x3) (In (ap x3 0) x0) leaving 2 subgoals.
The subproof is completed by applying L2.
Let x4 of type ι be given.
Assume H4:
∃ x5 . and (In x5 (x2 x4)) (x3 = setsum x4 x5).
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x5 . In x5 (x2 x4),
λ x5 . x3 = setsum x4 x5,
and (setsum_p x3) (In (ap x3 0) x0) leaving 2 subgoals.
The subproof is completed by applying H4.
Let x5 of type ι be given.
Assume H5:
In x5 (x2 x4).
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
setsum_p x3,
In (ap x3 0) x0 leaving 2 subgoals.
Apply H6 with
λ x6 x7 . setsum_p x7.
The subproof is completed by applying unknownprop_f61ccefc6bc57eb6c116b3bc3f27a552fe11c91770c4e9cfa989285bab91c3f5 with x4, x5.
Apply H6 with
λ x6 x7 . In (ap x7 0) x0.
Apply unknownprop_4ec3261e93b098949bcb767ddc18dfeab1e68ae68e2b47fdb55e525983415999 with
x4,
x5,
λ x6 x7 . In x7 x0.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Apply unknownprop_c5e2164052a280ad5b04f622e53815f0267ee33361e4345305e43303abef2c1b with
x0,
x2,
x3,
λ x4 x5 . In x5 (x1 x3) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H0 with
x3.
The subproof is completed by applying H1.