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.
Apply H0 with
λ x4 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . TwoRamseyGraph_4_5_24_ChurchNums_3x8 x4 x2 x1 x3 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) x4) x2 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) x1) x3 leaving 3 subgoals.
Apply H2 with
λ 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) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) x4 x1 x3 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) x4 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) x1) x3 leaving 8 subgoals.
Apply H1 with
λ x4 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) x4 x3 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) x4) x3 leaving 3 subgoals.
Apply H3 with
λ 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) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) x4 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) x4 leaving 8 subgoals.
Let x4 of type (ι → ι → ι) → (ι → ι → ι) → ο be given.
Assume H4:
x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5)).
The subproof is completed by applying H4.
Let x4 of type (ι → ι → ι) → (ι → ι → ι) → ο be given.
Assume H4:
x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x6)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x6)).
The subproof is completed by applying H4.
Let x4 of type (ι → ι → ι) → (ι → ι → ι) → ο be given.
Assume H4:
x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x7)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x7)).
The subproof is completed by applying H4.
Let x4 of type (ι → ι → ι) → (ι → ι → ι) → ο be given.
Assume H4:
x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x8)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) ((λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x8)).