Let x0 of type ι → (ι → ι → ι → ι) → ι → ι → ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Apply H0 with
λ x3 . λ x4 : ι → ι → ι . ∃ x5 : ι → ι → ι → ι . and (∀ x6 . x6 ∈ x3 ⟶ In_rec_G_iii x0 x6 (x5 x6)) (x4 = x0 x3 x5).
Let x3 of type ι be given.
Let x4 of type ι → ι → ι → ι be given.
Assume H1:
∀ x5 . x5 ∈ x3 ⟶ ∃ x6 : ι → ι → ι → ι . and (∀ x7 . x7 ∈ x5 ⟶ In_rec_G_iii x0 x7 (x6 x7)) (x4 x5 = x0 x5 x6).
Let x5 of type ο be given.
Assume H2:
∀ x6 : ι → ι → ι → ι . and (∀ x7 . x7 ∈ x3 ⟶ In_rec_G_iii x0 x7 (x6 x7)) (x0 x3 x4 = x0 x3 x6) ⟶ x5.
Apply H2 with
x4.
Apply andI with
∀ x6 . x6 ∈ x3 ⟶ In_rec_G_iii x0 x6 (x4 x6),
x0 x3 x4 = x0 x3 x4 leaving 2 subgoals.
Let x6 of type ι be given.
Assume H3: x6 ∈ x3.
Apply exandE_iiii with
λ x7 : ι → ι → ι → ι . ∀ x8 . x8 ∈ x6 ⟶ In_rec_G_iii x0 x8 (x7 x8),
λ x7 : ι → ι → ι → ι . x4 x6 = x0 x6 x7,
In_rec_G_iii x0 x6 (x4 x6) leaving 2 subgoals.
Apply H1 with
x6.
The subproof is completed by applying H3.
Let x7 of type ι → ι → ι → ι be given.
Assume H5: x4 x6 = x0 x6 x7.
Apply H5 with
λ x8 x9 : ι → ι → ι . In_rec_G_iii x0 x6 x9.
Apply In_rec_G_iii_c with
x0,
x6,
x7.
The subproof is completed by applying H4.
Let x6 of type (ι → ι → ι) → (ι → ι → ι) → ο be given.
Assume H3: x6 (x0 x3 x4) (x0 x3 x4).
The subproof is completed by applying H3.