Let x0 of type ι be given.
Assume H0:
∀ x1 . x1 ∈ x0 ⟶ SNo x1.
Assume H2: x0 = 0 ⟶ ∀ x1 : ο . x1.
Apply finite_max_exists with
{minus_SNo x1|x1 ∈ x0},
∃ x1 . SNo_min_of x0 x1 leaving 4 subgoals.
Let x1 of type ι be given.
Apply ReplE_impred with
x0,
λ x2 . minus_SNo x2,
x1,
SNo x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H4: x2 ∈ x0.
Apply H5 with
λ x3 x4 . SNo x4.
Apply SNo_minus_SNo with
x2.
Apply H0 with
x2.
The subproof is completed by applying H4.
Apply H1 with
finite {minus_SNo x1|x1 ∈ x0}.
Let x1 of type ι be given.
Apply H3 with
finite {minus_SNo x2|x2 ∈ x0}.
Let x2 of type ο be given.
Apply H6 with
x1.
Apply andI with
x1 ∈ omega,
equip {minus_SNo x3|x3 ∈ x0} x1 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply equip_tra with
{minus_SNo x3|x3 ∈ x0},
x0,
x1 leaving 2 subgoals.
Apply equip_sym with
x0,
{minus_SNo x3|x3 ∈ x0}.
Let x3 of type ο be given.
Assume H7:
∀ x4 : ι → ι . bij x0 {minus_SNo x5|x5 ∈ x0} x4 ⟶ x3.
Apply H7 with
minus_SNo.
Apply bijI with
x0,
{minus_SNo x4|x4 ∈ x0},
minus_SNo leaving 3 subgoals.
The subproof is completed by applying ReplI with
x0,
minus_SNo.
Let x4 of type ι be given.
Assume H8: x4 ∈ x0.
Let x5 of type ι be given.
Assume H9: x5 ∈ x0.
Let x6 of type ι → ι → ο be given.
set y7 to be λ x7 . x6 x7 x5 ⟶ x6 x5 x7
Claim L11: y7 x5
Assume H11: y7 x6 x6.
The subproof is completed by applying H11.
Apply minus_SNo_invol with
x4,
λ x8 x9 . (λ x10 . y7) x9 x8 leaving 2 subgoals.
Apply H0 with
x4.
The subproof is completed by applying H8.
Let x9 of type ι → ο be given.
The subproof is completed by applying H10 with
λ x10 x11 . (λ x12 . x9) (minus_SNo x10) (minus_SNo x11).
set y9 to be λ x9 . y8
Apply L12 with
λ x10 . y9 x10 y8 ⟶ y9 y8 x10 leaving 2 subgoals.
Assume H13: y9 y8 y8.
The subproof is completed by applying H13.
Apply minus_SNo_invol with
y7,
λ x10 . y9 leaving 2 subgoals.
Apply H1 with
y7.
The subproof is completed by applying H10.
The subproof is completed by applying L12.
Let x4 of type ι be given.
Apply ReplE_impred with
x0,
λ x5 . minus_SNo x5,
x4,
∃ x5 . and (x5 ∈ x0) (minus_SNo x5 = x4) leaving 2 subgoals.
The subproof is completed by applying H8.
Let x5 of type ι be given.
Assume H9: x5 ∈ x0.
Let x6 of type ο be given.
Apply H11 with
x5.
Apply andI with
x5 ∈ x0,
minus_SNo x5 = x4 leaving 2 subgoals.
The subproof is completed by applying H9.
Let x7 of type ι → ι → ο be given.
The subproof is completed by applying H10 with λ x8 x9 . x7 x9 x8.
The subproof is completed by applying H5.