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 H0: ChurchNum_3ary_proj_p x0.

Assume H1: ChurchNum_3ary_proj_p x1.

Assume H2: ChurchNum_8ary_proj_p x2.

Assume H3: ChurchNum_8ary_proj_p x3.

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 (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 x4 x2) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 x2) (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 x1 x3) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 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 (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) x4) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 x4) (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 x1 x3) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 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 (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5)) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5)) (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 x4 x3) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 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 (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5)) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5)) (ChurchNums_3x8_3_lt1_swap_1_2_ge1_rot2 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) x4) (ChurchNums_8_perm_0_7_6_5_4_3_2_1 x4) leaving 8 subgoals.