Let x0 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
Let x1 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
Apply H1 with
λ x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) = x2 ⟶ ∀ x3 : ο . x3) ⟶ ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x2 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) x2 = λ x3 x4 . x3) ⟶ (TwoRamseyGraph_3_5_Church13 x2 x1 = λ x3 x4 . x3) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) x1 = λ x3 x4 . x3) ⟶ False leaving 13 subgoals.
Assume H3: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) ⟶ ∀ x2 : ο . x2.
Apply FalseE with
((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = x1 ⟶ ∀ x2 : ο . x2) ⟶ ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = x1 ⟶ ∀ x2 : ο . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) x1 = λ x2 x3 . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) x1 = λ x2 x3 . x2) ⟶ False.
Apply H3.
Let x2 of type (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο be given.
Assume H4: x2 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3).
The subproof is completed by applying H4.
Apply H2 with
λ x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . (((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) = λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x4) ⟶ ∀ x3 : ο . x3) ⟶ ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) = x2 ⟶ ∀ x3 : ο . x3) ⟶ ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x4) = x2 ⟶ ∀ x3 : ο . x3) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x4) = λ x3 x4 . x3) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x4) x2 = λ x3 x4 . x3) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) x2 = λ x3 x4 . x3) ⟶ False leaving 13 subgoals.
Assume H3: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) ⟶ ∀ x2 : ο . x2.
Assume H4: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) ⟶ ∀ x2 : ο . x2.
Apply FalseE with
(((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) ⟶ ∀ x2 : ο . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) = λ x2 x3 . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 . x2) ⟶ False.
Apply H4.
Let x2 of type (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο be given.
Assume H5: x2 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x3).
The subproof is completed by applying H5.
Assume H3: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) ⟶ ∀ x2 : ο . x2.
Assume H4: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) ⟶ ∀ x2 : ο . x2.
Assume H5: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) ⟶ ∀ x2 : ο . x2.
Apply FalseE with
(TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) = λ x2 x3 . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) = λ x2 x3 . x2) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) (λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) = λ x2 x3 . x2) ⟶ False.
Apply H5.
Let x2 of type (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο be given.
Assume H6: x2 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x4) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x4).
The subproof is completed by applying H6.
Assume H3: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x3) ⟶ ∀ x2 : ο . x2.
Assume H4: ((λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x2) = λ x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 . x4) ⟶ ∀ x2 : ο . x2.