Let x0 of type ι → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Let x3 of type ο be given.
Assume H0:
∀ x4 . ordinal x4 ⟶ ∀ x5 : ι → ο . x0 x4 x5 ⟶ PNoLe x4 x5 x1 x2 ⟶ x3.
Apply unknownprop_d1cbc2e638db87d9196ef916ca16d68e66023c5a2d300e15ae355b0266749835 with
λ x4 x5 : (ι → (ι → ο) → ο) → ι → (ι → ο) → ο . x5 x0 x1 x2 ⟶ x3.
Assume H1:
(λ x4 : ι → (ι → ο) → ο . λ x5 . λ x6 : ι → ο . ∃ x7 . and (ordinal x7) (∃ x8 : ι → ο . and (x4 x7 x8) (PNoLe x7 x8 x5 x6))) x0 x1 x2.
Apply H1 with
x3.
Let x4 of type ι be given.
Assume H2:
(λ x5 . and (ordinal x5) (∃ x6 : ι → ο . and (x0 x5 x6) (PNoLe x5 x6 x1 x2))) x4.
Apply andE with
ordinal x4,
∃ x5 : ι → ο . and (x0 x4 x5) (PNoLe x4 x5 x1 x2),
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4:
∃ x5 : ι → ο . and (x0 x4 x5) (PNoLe x4 x5 x1 x2).
Apply H4 with
x3.
Let x5 of type ι → ο be given.
Assume H5:
(λ x6 : ι → ο . and (x0 x4 x6) (PNoLe x4 x6 x1 x2)) x5.
Apply andE with
x0 x4 x5,
PNoLe x4 x5 x1 x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H6: x0 x4 x5.
Assume H7:
PNoLe x4 x5 x1 x2.
Apply H0 with
x4,
x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
The subproof is completed by applying H7.