Let x0 of type ι → ι → ι → ι → 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 → ι → ι → ι → ι be given.
Let x2 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
Apply H1 with
λ x3 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . ((λ x4 x5 . x4) = λ x4 x5 . x3 x5 x5 x5 x5 x5 x5 x5 x5 x5 x4 x4 x4 x4 x4 x4 x4 x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x4) = x3 ⟶ ∀ x4 : ο . x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x4) = x1 ⟶ ∀ x4 : ο . x4) ⟶ ((λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x4) = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x3 = x1 ⟶ ∀ x4 : ο . x4) ⟶ (x3 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x4) x3 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x4) x1 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x4) x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_4_4_Church17 x3 x1 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_4_4_Church17 x3 x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_4_4_Church17 x1 x2 = λ x4 x5 . x5) ⟶ False leaving 17 subgoals.
Assume H4: (λ x3 x4 . x3) = λ x3 x4 . (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 . x5) x4 x4 x4 x4 x4 x4 x4 x4 x4 x3 x3 x3 x3 x3 x3 x3 x3.
Apply FalseE with
(((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) = λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) ⟶ ∀ x3 : ο . x3) ⟶ ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) = x1 ⟶ ∀ x3 : ο . x3) ⟶ ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) = x2 ⟶ ∀ x3 : ο . x3) ⟶ ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) = x1 ⟶ ∀ x3 : ο . x3) ⟶ ((λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) x1 = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) x2 = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) x1 = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_4_4_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) x2 = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_4_4_Church17 x1 x2 = λ x3 x4 . x4) ⟶ False.
Apply L0.
The subproof is completed by applying H4.
Assume H4: (λ x3 x4 . x3) = λ x3 x4 . (λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 . x6) x4 x4 x4 x4 x4 x4 x4 x4 x4 x3 x3 x3 x3 x3 x3 x3 x3.