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) → ((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 x2 : ((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 . (λ x3 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 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 . λ x5 x6 x7 : (ι → ι) → ι → ι . x3 (x4 x5 x7 x7 x7 x7 x7 x7 x7) (x4 x7 x6 x6 x6 x6 x6 x6 x6) (x4 x6 x5 x5 x5 x5 x5 x5 x5)) ((λ x3 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 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 . λ x5 x6 x7 : (ι → ι) → ι → ι . x3 (x4 x5 x7 x7 x7 x7 x7 x7 x7) (x4 x7 x6 x6 x6 x6 x6 x6 x6) (x4 x6 x5 x5 x5 x5 x5 x5 x5)) x0 x2) ((λ 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) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x3 x4 x11 x10 x9 x8 x7 x6 x5) x2) = x0 leaving 8 subgoals.
Let x2 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.
Assume H1: x2 ((λ x3 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 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 . λ x5 x6 x7 : (ι → ι) → ι → ι . x3 (x4 x5 x7 x7 x7 x7 x7 x7 x7) (x4 x7 x6 x6 x6 x6 x6 x6 x6) (x4 x6 x5 x5 x5 x5 x5 x5 x5)) ((λ x3 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 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 . λ x5 x6 x7 : (ι → ι) → ι → ι . x3 (x4 x5 x7 x7 x7 x7 x7 x7 x7) (x4 x7 x6 x6 x6 x6 x6 x6 x6) (x4 x6 x5 x5 x5 x5 x5 x5 x5)) x0 (λ x3 x4 x5 x6 x7 x8 x9 x10 : (ι → ι) → ι → ι . x3)) ((λ 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) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x3 x4 x11 x10 x9 x8 x7 x6 x5) (λ x3 x4 x5 x6 x7 x8 x9 x10 : (ι → ι) → ι → ι . x3))) x0.
The subproof is completed by applying H1.
Let x2 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.
Assume H1: x2 ((λ x3 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 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 . λ x5 x6 x7 : (ι → ι) → ι → ι . x3 (x4 x5 x7 x7 x7 x7 x7 x7 x7) (x4 x7 x6 x6 x6 x6 x6 x6 x6) (x4 x6 x5 x5 x5 x5 x5 x5 x5)) ((λ x3 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 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 . λ x5 x6 x7 : (ι → ι) → ι → ι . x3 (x4 x5 x7 x7 x7 x7 x7 x7 x7) (x4 x7 x6 x6 x6 x6 x6 x6 x6) (x4 x6 x5 x5 x5 x5 x5 x5 x5)) x0 (λ x3 x4 x5 x6 x7 x8 x9 x10 : (ι → ι) → ι → ι . x4)) ((λ 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) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x4 x5 x6 x7 x8 x9 x10 x11 : (ι → ι) → ι → ι . x3 x4 x11 x10 x9 x8 x7 x6 x5) (λ x3 x4 x5 x6 x7 x8 x9 x10 : (ι → ι) → ι → ι . x4))) x0.
The subproof is completed by applying H1.
Let x2 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.
Assume H1: x2 ((λ x3 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 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 . ...) ... ...) ....