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.
Apply andI with
x0 = x2,
x1 = x3 leaving 2 subgoals.
Apply proj0_pair_eq with
x0,
x1,
λ x4 x5 . x4 = x2.
Apply proj0_pair_eq with
x2,
x3,
λ x4 x5 . proj0 (setsum x0 x1) = x4.
Apply H0 with
λ x4 x5 . proj0 x5 = proj0 (setsum x2 x3).
Let x4 of type ι → ι → ο be given.
The subproof is completed by applying H1.
Apply proj1_pair_eq with
x0,
x1,
λ x4 x5 . x4 = x3.
Apply proj1_pair_eq with
x2,
x3,
λ x4 x5 . proj1 (setsum x0 x1) = x4.
Apply H0 with
λ x4 x5 . proj1 x5 = proj1 (setsum x2 x3).
Let x4 of type ι → ι → ο be given.
The subproof is completed by applying H1.