Let x0 of type ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (ι → ι) be given.
Let x1 of type ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (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) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (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) → CN (ι → ι) be given.
Assume H1: ∀ x4 : ο . ((x0 = λ x5 x6 x7 : (ι → ι) → ι → ι . x5) ⟶ (x2 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x9) ⟶ x4) ⟶ ((x0 = λ x5 x6 x7 : (ι → ι) → ι → ι . x6) ⟶ (x2 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x6) ⟶ x4) ⟶ ((x0 = λ x5 x6 x7 : (ι → ι) → ι → ι . x6) ⟶ (x2 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x12) ⟶ x4) ⟶ ((x0 = λ x5 x6 x7 : (ι → ι) → ι → ι . x7) ⟶ (x2 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x9) ⟶ x4) ⟶ x4.
Assume H2: ∀ x4 : ο . ((x1 = λ x5 x6 x7 : (ι → ι) → ι → ι . x5) ⟶ (x3 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x9) ⟶ x4) ⟶ ((x1 = λ x5 x6 x7 : (ι → ι) → ι → ι . x6) ⟶ (x3 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x6) ⟶ x4) ⟶ ((x1 = λ x5 x6 x7 : (ι → ι) → ι → ι . x6) ⟶ (x3 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x12) ⟶ x4) ⟶ ((x1 = λ x5 x6 x7 : (ι → ι) → ι → ι . x7) ⟶ (x3 = λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x9) ⟶ x4) ⟶ x4.
Apply H1 with
(TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) (λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8) x0 x2 = λ x4 x5 . x4) ⟶ (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) (λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8) x1 x3 = λ x4 x5 . x4) ⟶ ChurchNums_3x8_neq (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) (λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8) x0 x2 ⟶ ChurchNums_3x8_neq (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) (λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8) x1 x3 ⟶ ChurchNums_3x8_neq x0 x2 x1 x3 ⟶ False leaving 4 subgoals.
Assume H3: x0 = λ x4 x5 x6 : (ι → ι) → ι → ι . x4.
Assume H4: x2 = λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8.
Assume H7:
ChurchNums_3x8_neq (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) (λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8) x0 x2.
Apply FalseE with
ChurchNums_3x8_neq (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) (λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8) x1 x3 ⟶ ChurchNums_3x8_neq x0 x2 x1 x3 ⟶ False.
Apply H7.
Apply unknownprop_489a19599530946830ae79502aec6ef7b2f064765691a3ca83405abd2ab867f4 with
λ x4 x5 x6 : (ι → ι) → ι → ι . x4,
x0,
λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x8,
x2 leaving 2 subgoals.
Let x4 of type (((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (ι → ι)) → (((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (ι → ι)) → ο be given.
The subproof is completed by applying H3 with λ x5 x6 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x4 x6 x5.
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) → CN (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) → ι → ι) → CN (ι → ι)) → ο be given.
The subproof is completed by applying H4 with λ x5 x6 : ((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) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x4 x6 x5.
Assume H3: x0 = λ x4 x5 x6 : (ι → ι) → ι → ι . x5.
Assume H4: x2 = λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x5.
Apply H3 with
λ x4 x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x6 x7 x8 : (ι → ι) → ι → ι . x6) (λ x6 x7 x8 x9 x10 x11 x12 x13 : (ι → ι) → ι → ι . x10) x5 x2 = λ x6 x7 . x6) ⟶ (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x6 x7 x8 : (ι → ι) → ι → ι . x6) (λ x6 x7 x8 x9 x10 x11 x12 x13 : (ι → ι) → ι → ι . x10) x1 x3 = λ x6 x7 . x6) ⟶ ChurchNums_3x8_neq (λ x6 x7 x8 : (ι → ι) → ι → ι . x6) (λ x6 x7 x8 x9 x10 x11 x12 x13 : (ι → ι) → ι → ι . x10) x5 x2 ⟶ ChurchNums_3x8_neq (λ x6 x7 x8 : (ι → ι) → ι → ι . x6) (λ x6 x7 x8 x9 x10 x11 x12 x13 : (ι → ι) → ι → ι . x10) x1 x3 ⟶ ChurchNums_3x8_neq x5 x2 x1 x3 ⟶ False.
Apply H4 with
λ x4 x5 : ((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) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . ... ⟶ TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x6 x7 x8 : (ι → ι) → ι → ι . x6) ... ... ... = ... ⟶ ChurchNums_3x8_neq (λ x6 x7 x8 : (ι → ι) → ι → ι . x6) (λ x6 x7 x8 x9 x10 x11 x12 x13 : (ι → ι) → ι → ι . x10) (λ x6 x7 x8 : (ι → ι) → ι → ι . x7) x5 ⟶ ChurchNums_3x8_neq (λ x6 x7 x8 : (ι → ι) → ι → ι . x6) (λ x6 x7 x8 x9 x10 x11 x12 x13 : (ι → ι) → ι → ι . x10) x1 x3 ⟶ ChurchNums_3x8_neq (λ x6 x7 x8 : (ι → ι) → ι → ι . x7) x5 x1 x3 ⟶ False.