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.
Let x4 of type ι be given.
Apply unknownprop_2dc9e26456bb41cf29ae85d63c3e8887dbdb5a0601149d8bd58cb1df95ffb8a5 with
x0,
λ x5 . struct_comp x5 x1 x2 x3 x4 = 0 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x5 of type ι be given.
Let x6 of type ι → ι → ο be given.
Assume H1: x5 = 0.
Apply pack_r_0_eq2 with
x5,
x6,
λ x7 x8 . lam x7 (λ x9 . ap x3 (ap x4 x9)) = 0.
Apply H1 with
λ x7 x8 . lam x8 (λ x9 . ap x3 (ap x4 x9)) = 0.
Apply Empty_eq with
lam 0 (λ x7 . ap x3 (ap x4 x7)).
Let x7 of type ι be given.
Assume H2:
x7 ∈ lam 0 (λ x8 . ap x3 (ap x4 x8)).
Apply lamE with
0,
λ x8 . ap x3 (ap x4 x8),
x7,
False leaving 2 subgoals.
The subproof is completed by applying H2.
Let x8 of type ι be given.
Assume H3:
(λ x9 . and (x9 ∈ 0) (∃ x10 . and (x10 ∈ ap x3 (ap x4 x9)) (x7 = setsum x9 x10))) x8.
Apply H3 with
False.
Assume H4: x8 ∈ 0.
Apply FalseE with
(∃ x9 . and (x9 ∈ ap x3 (ap x4 x8)) (x7 = setsum x8 x9)) ⟶ False.
Apply EmptyE with
x8.
The subproof is completed by applying H4.