Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Assume H1: x2 0.
Assume H2:
∀ x3 . In x3 x0 ⟶ x2 (Inj1 x3).
Apply unknownprop_eb8e8f72a91f1b934993d4cb19c84c8270f73a3626f3022b683d960a7fef89cb with
x1 = 0,
∃ x3 . and (In x3 x0) (x1 = Inj1 x3),
x2 x1 leaving 3 subgoals.
Apply unknownprop_46cd588b09db5f692c112630dcd23fdd67e6834d19385c7fc3543b49426246b4 with
x0,
x1.
The subproof is completed by applying H0.
Assume H3: x1 = 0.
Apply H3 with
λ x3 x4 . x2 x4.
The subproof is completed by applying H1.
Assume H3:
∃ x3 . and (In x3 x0) (x1 = Inj1 x3).
Apply H3 with
x2 x1.
Let x3 of type ι be given.
Assume H4:
(λ x4 . and (In x4 x0) (x1 = Inj1 x4)) x3.
Apply andE with
In x3 x0,
x1 = Inj1 x3,
x2 x1 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H6 with
λ x4 x5 . x2 x5.
Apply H2 with
x3.
The subproof is completed by applying H5.