Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply unknownprop_2c926b240fc658005337215abfdc8124a6f6eea17ba9f4df80d254bab3845972 with
λ x3 x4 : ι → ι → ο . x4 3 (lam 3 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 x2))).
Let x3 of type ι be given.
Assume H0:
In x3 (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))).
Apply unknownprop_f25818182af6b093121a8b5d43847162c8ea91396e524cca02557613a430a57a with
3,
λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2),
x3.
The subproof is completed by applying H0.
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x4 . In x4 3,
λ x4 . ∃ x5 . and (In x5 (If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) (x3 = setsum x4 x5),
∃ x4 . and (In x4 3) (∃ x5 . x3 = setsum x4 x5) leaving 2 subgoals.
The subproof is completed by applying L1.
Let x4 of type ι be given.
Apply unknownprop_3848cfb1fd522cb609408da39f227a9c05924a24919f21041d0880590b824ef5 with
λ x5 . In x5 (If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)),
λ x5 . x3 = setsum x4 x5,
∃ x5 . and (In x5 3) (∃ x6 . x3 = setsum x5 x6) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι be given.
Assume H4:
In x5 (If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)).
Let x6 of type ο be given.
Assume H6:
∀ x7 . and (In x7 3) (∃ x8 . x3 = setsum x7 x8) ⟶ x6.
Apply H6 with
x4.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
In x4 3,
∃ x7 . x3 = setsum x4 x7 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x7 of type ο be given.
Assume H7:
∀ x8 . x3 = setsum x4 x8 ⟶ x7.
Apply H7 with
x5.
The subproof is completed by applying H5.