Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0.
Let x2 of type ι be given.
Let x3 of type ι → ι → ι be given.
Assume H2: ∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2.
Let x4 of type ι → ι → ι be given.
Assume H4:
∀ x5 . x5 ∈ setprod x0 x2 ⟶ ∀ x6 . x6 ∈ setprod x0 x2 ⟶ lam 2 (λ x7 . If_i (x7 = 0) (x1 (ap x5 0) (ap x6 0)) (x3 (ap x5 1) (ap x6 1))) = x4 x5 x6.
Apply H1 with
explicit_Group (setprod x0 x2) x4.
Assume H5:
and (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 ∈ x0) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x1 x5 (x1 x6 x7) = x1 (x1 x5 x6) x7).
Apply H5 with
(∃ x5 . and (x5 ∈ x0) (and (∀ x6 . x6 ∈ x0 ⟶ and (x1 x5 x6 = x6) (x1 x6 x5 = x6)) (∀ x6 . x6 ∈ x0 ⟶ ∃ x7 . and (x7 ∈ x0) (and (x1 x6 x7 = x5) (x1 x7 x6 = x5))))) ⟶ explicit_Group (setprod x0 x2) x4.
Assume H6: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 ∈ x0.
Assume H7: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x1 x5 (x1 x6 x7) = x1 (x1 x5 x6) x7.
Assume H8:
∃ x5 . and (x5 ∈ x0) (and (∀ x6 . x6 ∈ x0 ⟶ and (x1 x5 x6 = x6) (x1 x6 x5 = x6)) (∀ x6 . x6 ∈ x0 ⟶ ∃ x7 . and (x7 ∈ x0) (and (x1 x6 x7 = x5) (x1 x7 x6 = x5)))).
Apply H3 with
explicit_Group (setprod x0 x2) x4.
Assume H9:
and (∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x5 x6 ∈ x2) (∀ x5 . ... ⟶ ∀ x6 . ... ⟶ ∀ x7 . ... ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) ...).