Let x0 of type ο be given.
Assume H0:
∀ x1 : ι → ι . bij 4 (prim4 2) x1 ⟶ x0.
Apply H0 with
λ x1 . ap (lam 4 (λ x2 . If_i (x2 = 0) 0 (If_i (x2 = 1) 1 (If_i (x2 = 2) (Sing 1) 2)))) x1.
Apply bijI with
4,
prim4 2,
λ x1 . ap (lam 4 (λ x2 . If_i (x2 = 0) 0 (If_i (x2 = 1) 1 (If_i (x2 = 2) (Sing 1) 2)))) x1 leaving 3 subgoals.
Let x1 of type ι be given.
Assume H1: x1 ∈ 4.
Apply cases_4 with
x1,
λ x2 . (λ x3 . ap (lam 4 (λ x4 . If_i (x4 = 0) 0 (If_i (x4 = 1) 1 (If_i (x4 = 2) (Sing 1) 2)))) x3) x2 ∈ prim4 2 leaving 5 subgoals.
The subproof is completed by applying H1.
Apply tuple_4_0_eq with
0,
1,
Sing 1,
2,
λ x2 x3 . x3 ∈ prim4 2.
The subproof is completed by applying Empty_In_Power with 2.
Apply tuple_4_1_eq with
0,
1,
Sing 1,
2,
λ x2 x3 . x3 ∈ prim4 2.
Apply PowerI with
2,
1.
The subproof is completed by applying Subq_1_2.
Apply tuple_4_2_eq with
0,
1,
Sing 1,
2,
λ x2 x3 . x3 ∈ prim4 2.
Apply PowerI with
2,
Sing 1.
Let x2 of type ι be given.
Apply SingE with
1,
x2,
λ x3 x4 . x4 ∈ 2 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying In_1_2.
Apply tuple_4_3_eq with
0,
1,
Sing 1,
2,
λ x2 x3 . x3 ∈ prim4 2.
The subproof is completed by applying Self_In_Power with 2.
Let x1 of type ι be given.
Assume H1: x1 ∈ 4.
Let x2 of type ι be given.
Assume H2: x2 ∈ 4.
Apply cases_4 with
x1,
λ x3 . (λ x4 . ap (lam 4 (λ x5 . If_i (x5 = 0) 0 (If_i (x5 = 1) 1 (If_i (x5 = 2) (Sing 1) 2)))) x4) x3 = (λ x4 . ap (lam 4 (λ x5 . If_i (x5 = 0) 0 (If_i (x5 = 1) 1 (If_i (x5 = 2) (Sing 1) 2)))) x4) x2 ⟶ x3 = x2 leaving 5 subgoals.
The subproof is completed by applying H1.
Apply cases_4 with
x2,
λ x3 . (λ x4 . ap (lam 4 (λ x5 . If_i (x5 = 0) 0 (If_i (x5 = 1) 1 (If_i (x5 = 2) (Sing 1) 2)))) x4) 0 = (λ x4 . ap (lam 4 (λ x5 . If_i (x5 = 0) 0 (If_i (x5 = 1) 1 (If_i (x5 = 2) (Sing 1) 2)))) x4) x3 ⟶ 0 = x3 leaving 5 subgoals.
The subproof is completed by applying H2.
Assume H3:
(λ x3 . ap (lam 4 (λ x4 . If_i (x4 = 0) 0 (If_i (... = 1) 1 ...))) ...) 0 = ....