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.
Let x2 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
Let x3 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
Apply H1 with
λ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . ((λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) = x4 ⟶ ∀ x5 : ο . x5) ⟶ ((λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) = x1 ⟶ ∀ x5 : ο . x5) ⟶ ((λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) = x2 ⟶ ∀ x5 : ο . x5) ⟶ ((λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x4 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (x4 = x2 ⟶ ∀ x5 : ο . x5) ⟶ (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x1 = x2 ⟶ ∀ x5 : ο . x5) ⟶ (x1 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) x4 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) x1 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) x2 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) x3 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 x4 x1 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 x4 x2 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 x4 x3 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 x1 x2 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 x1 x3 = λ x5 x6 . x6) ⟶ (TwoRamseyGraph_3_5_Church13 x2 x3 = λ x5 x6 . x6) ⟶ False leaving 13 subgoals.
Assume H5: ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) ⟶ ∀ x4 : ο . x4.
Apply FalseE with
((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = x1 ⟶ ∀ x4 : ο . x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = x2 ⟶ ∀ x4 : ο . x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = x3 ⟶ ∀ x4 : ο . x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = x1 ⟶ ∀ x4 : ο . x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = x2 ⟶ ∀ x4 : ο . x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) x1 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) x3 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) x1 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) x3 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 x1 x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 x1 x3 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_5_Church13 x2 x3 = λ x4 x5 . x5) ⟶ False.
Apply H5.
Let x4 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: x4 (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5).
The subproof is completed by applying H6.
Assume H5: ... ⟶ ∀ x4 : ο . x4.