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 → ο be given.
Assume H1:
∀ x4 . In x4 x0 ⟶ x3 (setsum 0 x4).
Assume H2:
∀ x4 . In x4 x1 ⟶ x3 (setsum 1 x4).
Apply unknownprop_eb8e8f72a91f1b934993d4cb19c84c8270f73a3626f3022b683d960a7fef89cb with
∃ x4 . and (In x4 x0) (x2 = setsum 0 x4),
∃ x4 . and (In x4 x1) (x2 = setsum 1 x4),
x3 x2 leaving 3 subgoals.
Apply unknownprop_c529973cb32f8d02a3950eda53e547d40c4a0e8faca1777353233a3377534f09 with
x0,
x1,
x2.
The subproof is completed by applying H0.
Apply H3 with
x3 x2.
Let x4 of type ι be given.
Assume H4:
(λ x5 . and (In x5 x0) (x2 = setsum 0 x5)) x4.
Apply andE with
In x4 x0,
x2 = setsum 0 x4,
x3 x2 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H6 with
λ x5 x6 . x3 x6.
Apply H1 with
x4.
The subproof is completed by applying H5.
Apply H3 with
x3 x2.
Let x4 of type ι be given.
Assume H4:
(λ x5 . and (In x5 x1) (x2 = setsum 1 x5)) x4.
Apply andE with
In x4 x1,
x2 = setsum 1 x4,
x3 x2 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H6 with
λ x5 x6 . x3 x6.
Apply H2 with
x4.
The subproof is completed by applying H5.