Claim L0: ...

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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 . (TwoRamseyGraph_3_6_Church17 x3 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x12) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x3 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x14) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x1 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x12) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x1 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x14) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x2 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x12) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x2 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x14) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x3 x1 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x3 x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x1 x2 = λ x4 x5 . x5) ⟶ False leaving 8 subgoals.

Assume H4: TwoRamseyGraph_3_6_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 . x11) = λ x3 x4 . x4.

Assume H5: TwoRamseyGraph_3_6_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 . x13) = λ x3 x4 . x4.

Apply FalseE with (TwoRamseyGraph_3_6_Church17 x1 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x11) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x1 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x13) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x2 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x11) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 x2 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x13) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_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_3_6_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_3_6_Church17 x1 x2 = λ x3 x4 . x4) ⟶ False.

Apply L0.

The subproof is completed by applying H5.

Apply H2 with λ x3 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . (TwoRamseyGraph_3_6_Church17 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x5) (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x12) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x5) (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x14) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x3 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x12) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x3 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x14) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x2 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x12) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x2 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x14) = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x5) x3 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 . x5) x2 = λ x4 x5 . x5) ⟶ (TwoRamseyGraph_3_6_Church17 x3 x2 = λ x4 x5 . x5) ⟶ False leaving 8 subgoals.

Assume H4: TwoRamseyGraph_3_6_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x4) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x11) = λ x3 x4 . x4.

Assume H5: TwoRamseyGraph_3_6_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x4) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x13) = λ x3 x4 . x4.

Assume H6: TwoRamseyGraph_3_6_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 . x11) = λ x3 x4 . x4.

Assume H7: TwoRamseyGraph_3_6_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 . x13) = λ x3 x4 . x4.

Apply FalseE with ... ⟶ TwoRamseyGraph_3_6_Church17 x2 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x13) = ... ⟶ (TwoRamseyGraph_3_6_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x4) (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x3) = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_Church17 (λ x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 . x4) x2 = λ x3 x4 . x4) ⟶ (TwoRamseyGraph_3_6_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) ⟶ False.

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Proofgold Proof