Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Apply H0 with
equip (setprod x0 x2) (setprod x1 x3).
Let x4 of type ι → ι be given.
Apply bijE with
x0,
x1,
x4,
equip (setprod x0 x2) (setprod x1 x3) leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3: ∀ x5 . x5 ∈ x0 ⟶ x4 x5 ∈ x1.
Assume H4: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 = x4 x6 ⟶ x5 = x6.
Assume H5:
∀ x5 . x5 ∈ x1 ⟶ ∃ x6 . and (x6 ∈ x0) (x4 x6 = x5).
Apply H1 with
equip (setprod x0 x2) (setprod x1 x3).
Let x5 of type ι → ι be given.
Apply bijE with
x2,
x3,
x5,
equip (setprod x0 x2) (setprod x1 x3) leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H7: ∀ x6 . x6 ∈ x2 ⟶ x5 x6 ∈ x3.
Assume H8: ∀ x6 . x6 ∈ x2 ⟶ ∀ x7 . x7 ∈ x2 ⟶ x5 x6 = x5 x7 ⟶ x6 = x7.
Assume H9:
∀ x6 . x6 ∈ x3 ⟶ ∃ x7 . and (x7 ∈ x2) (x5 x7 = x6).
Let x6 of type ο be given.
Apply H10 with
λ x7 . lam 2 (λ x8 . If_i (x8 = 0) (x4 (ap x7 0)) (x5 (ap x7 1))).
Apply bijI with
setprod x0 x2,
setprod x1 x3,
λ x7 . lam 2 (λ x8 . If_i (x8 = 0) (x4 (ap x7 0)) (x5 (ap x7 1))) leaving 3 subgoals.
Let x7 of type ι be given.
Apply tuple_2_setprod with
x1,
x3,
x4 (ap x7 0),
x5 (ap x7 1) leaving 2 subgoals.
Apply H3 with
ap x7 0.
Apply ap0_Sigma with
x0,
λ x8 . x2,
x7.
The subproof is completed by applying H11.
Apply H7 with
ap x7 1.
Apply ap1_Sigma with
x0,
λ x8 . x2,
x7.
The subproof is completed by applying H11.
Let x7 of type ι be given.
Let x8 of type ι be given.
Assume H13:
lam 2 (λ x9 . If_i (x9 = 0) (x4 (ap x7 0)) (x5 (ap x7 1))) = lam 2 (λ x9 . If_i (x9 = 0) (x4 (ap x8 0)) (x5 (ap x8 1))).
Apply tuple_Sigma_eta with
x0,
λ x9 . x2,
x7,
λ x9 x10 . x9 = x8 leaving 2 subgoals.
The subproof is completed by applying H11.
Apply tuple_Sigma_eta with
x0,
λ x9 . x2,
x8,
λ x9 x10 . lam 2 (λ x11 . If_i (x11 = 0) ... ...) = ... leaving 2 subgoals.