Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Let x6 of type ι be given.
Assume H5: x6 ∈ x0.
Let x7 of type ι be given.
Assume H6: x7 ∈ x0.
Let x8 of type ι be given.
Assume H7: x8 ∈ x0.
Let x9 of type ι be given.
Assume H8: x9 ∈ x0.
Let x10 of type ι be given.
Assume H9: x10 ∈ x0.
Assume H10:
27706.. x1 x2 x3 x4 x5 x6 x7 x8 x9 x10.
Let x11 of type ο be given.
Assume H11:
83424.. x1 x2 x6 x7 x10 x3 x4 x8 x9 ⟶ (x2 = x5 ⟶ ∀ x12 : ο . x12) ⟶ (x6 = x5 ⟶ ∀ x12 : ο . x12) ⟶ (x7 = x5 ⟶ ∀ x12 : ο . x12) ⟶ (x10 = x5 ⟶ ∀ x12 : ο . x12) ⟶ (x3 = x5 ⟶ ∀ x12 : ο . x12) ⟶ (x4 = x5 ⟶ ∀ x12 : ο . x12) ⟶ (x8 = x5 ⟶ ∀ x12 : ο . x12) ⟶ (x9 = x5 ⟶ ∀ x12 : ο . x12) ⟶ not (x1 x2 x5) ⟶ x1 x6 x5 ⟶ x1 x7 x5 ⟶ not (x1 x10 x5) ⟶ not (x1 x3 x5) ⟶ not (x1 x4 x5) ⟶ x1 x8 x5 ⟶ x1 x9 x5 ⟶ x11.
Apply H10 with
x11.
Assume H13:
83424.. x1 x2 x3 x4 x5 x6 x7 x8 x9.
Apply H13 with
(x2 = x10 ⟶ ∀ x12 : ο . x12) ⟶ (x3 = x10 ⟶ ∀ x12 : ο . x12) ⟶ (x4 = x10 ⟶ ∀ x12 : ο . x12) ⟶ (x5 = x10 ⟶ ∀ x12 : ο . x12) ⟶ (x6 = x10 ⟶ ∀ x12 : ο . x12) ⟶ (x7 = x10 ⟶ ∀ x12 : ο . x12) ⟶ (x8 = x10 ⟶ ∀ x12 : ο . x12) ⟶ (x9 = x10 ⟶ ∀ x12 : ο . x12) ⟶ not (x1 x2 x10) ⟶ x1 x3 x10 ⟶ x1 x4 x10 ⟶ not (x1 x5 x10) ⟶ not (x1 x6 x10) ⟶ not (x1 x7 x10) ⟶ x1 x8 x10 ⟶ x1 x9 x10 ⟶ x11.
Assume H14:
3109c.. x1 x2 x3 x4 x5 x6 x7 x8.
Apply H14 with
... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ (... ⟶ ∀ x12 : ο . x12) ⟶ not (x1 x2 x10) ⟶ x1 x3 x10 ⟶ x1 x4 x10 ⟶ not (x1 x5 x10) ⟶ not (x1 x6 x10) ⟶ not (x1 x7 x10) ⟶ x1 x8 x10 ⟶ x1 x9 x10 ⟶ x11.