Let x0 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
Let x1 of type ο be given.
Apply H0 with
λ x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . (∀ x3 x4 : (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 . (∀ x5 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x5 ⟶ Church13_p (x3 x5)) ⟶ (∀ x5 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x5 ⟶ Church13_p (x4 x5)) ⟶ (∀ x5 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x3 (x4 x5) = x5) ⟶ (∀ x5 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x4 (x3 x5) = x5) ⟶ (∀ x5 x6 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x5 ⟶ Church13_p x6 ⟶ TwoRamseyGraph_3_5_Church13 x5 x6 = TwoRamseyGraph_3_5_Church13 (x3 x5) (x3 x6)) ⟶ (x3 x2 = λ x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 . x5) ⟶ x1) ⟶ x1 leaving 13 subgoals.
Assume H1:
∀ x2 x3 : (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 . (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x4 ⟶ Church13_p (x2 x4)) ⟶ (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x4 ⟶ Church13_p (x3 x4)) ⟶ (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x2 (x3 x4) = x4) ⟶ (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x3 (x2 x4) = x4) ⟶ (∀ x4 x5 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x4 ⟶ Church13_p x5 ⟶ TwoRamseyGraph_3_5_Church13 x4 x5 = TwoRamseyGraph_3_5_Church13 (x2 x4) (x2 x5)) ⟶ (x2 (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) ⟶ x1.
Apply H1 with
λ x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x2,
λ x2 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x2 leaving 6 subgoals.
Let x2 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
The subproof is completed by applying H2.
Let x2 of type ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι be given.
The subproof is completed by applying H2.
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) → (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο be given.
Assume H2: x3 ((λ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x4) x2) x2.
The subproof is completed by applying H2.
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) → (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο be given.
Assume H2: x3 ((λ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x4) x2) x2.
The subproof is completed by applying H2.
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.
Let x4 of type (ι → ι → ι) → (ι → ι → ι) → ο be given.
The subproof is completed by applying 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 H2: 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 H2.
Assume H1:
∀ x2 x3 : (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 . (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x4 ⟶ Church13_p (x2 x4)) ⟶ (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x4 ⟶ Church13_p (x3 x4)) ⟶ (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x2 (x3 x4) = x4) ⟶ (∀ x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x3 (x2 x4) = x4) ⟶ (∀ x4 x5 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . Church13_p x4 ⟶ Church13_p x5 ⟶ TwoRamseyGraph_3_5_Church13 x4 x5 = TwoRamseyGraph_3_5_Church13 (x2 x4) (x2 x5)) ⟶ (x2 (λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x5) = λ x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 . x4) ⟶ x1.
Apply H1 with
Church13_perm_12_0_1_2_3_4_5_6_7_8_9_10_11,
Church13_perm_1_2_3_4_5_6_7_8_9_10_11_12_0 leaving 6 subgoals.
The subproof is completed by applying unknownprop_b16b38441259a05332428d0f2db526e9dc2ef83fe954c8261623fb4dabc28885.
The subproof is completed by applying unknownprop_8fb3724bdbb3c5ecf278da23741ea89123ac92b2f2457be6608d21a3bd72d9fd.
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) → (ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο be given.
Apply L3 with
λ x3 x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . x4 = x2.
Apply L2 with
λ x3 x4 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . (λ x5 : ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι → ι . λ x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 . x5 x18 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17) x4 = x2.
Let x3 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: x3 x2 x2.
The subproof is completed by applying H4.