Let x0 of type ι → ι → ο be given.
Let x1 of type ι → ι be given.
Assume H0:
∀ x2 . x2 ∈ u16 ⟶ ∀ x3 . x3 ∈ u16 ⟶ x0 (x1 x2) (x1 x3) ⟶ x0 x2 x3.
Let x2 of type ι be given.
Assume H2:
∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ not (x0 x3 x4).
Let x3 of type ι be given.
Assume H3: x3 ∈ {x1 x4|x4 ∈ x2}.
Let x4 of type ι be given.
Assume H4: x4 ∈ {x1 x5|x5 ∈ x2}.
Apply ReplE_impred with
x2,
x1,
x3,
not (x0 x3 x4) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι be given.
Assume H5: x5 ∈ x2.
Assume H6: x3 = x1 x5.
Apply ReplE_impred with
x2,
x1,
x4,
not (x0 x3 x4) leaving 2 subgoals.
The subproof is completed by applying H4.
Let x6 of type ι be given.
Assume H7: x6 ∈ x2.
Assume H8: x4 = x1 x6.
Apply H6 with
λ x7 x8 . not (x0 x8 x4).
Apply H8 with
λ x7 x8 . not (x0 (x1 x5) x8).
Assume H9: x0 (x1 x5) (x1 x6).
Apply H2 with
x5,
x6 leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
Apply H0 with
x5,
x6 leaving 3 subgoals.
Apply H1 with
x5.
The subproof is completed by applying H5.
Apply H1 with
x6.
The subproof is completed by applying H7.
The subproof is completed by applying H9.