Let x0 of type ι be given.
Assume H0:
x0 ∈ omega.
Apply nat_ind with
λ x1 . add_SNo (minus_SNo x0) x1 ∈ int leaving 2 subgoals.
Apply add_SNo_0R with
minus_SNo x0,
λ x1 x2 . x2 ∈ int leaving 2 subgoals.
Apply SNo_minus_SNo with
x0.
Apply ordinal_SNo with
x0.
Apply nat_p_ordinal with
x0.
Apply omega_nat_p with
x0.
The subproof is completed by applying H0.
Apply int_minus_SNo_omega with
x0.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Apply ordinal_ordsucc_SNo_eq with
x1,
λ x2 x3 . add_SNo (minus_SNo x0) x3 ∈ int leaving 2 subgoals.
Apply nat_p_ordinal with
x1.
The subproof is completed by applying H1.
Apply add_SNo_com_3_0_1 with
minus_SNo x0,
1,
x1,
λ x2 x3 . x3 ∈ int leaving 4 subgoals.
Apply SNo_minus_SNo with
x0.
Apply ordinal_SNo with
x0.
Apply nat_p_ordinal with
x0.
Apply omega_nat_p with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_1.
Apply ordinal_SNo with
x1.
Apply nat_p_ordinal with
x1.
The subproof is completed by applying H1.
Apply int_SNo_cases with
λ x2 . add_SNo (minus_SNo x0) x1 = x2 ⟶ add_SNo 1 (add_SNo (minus_SNo x0) x1) ∈ int,
add_SNo (minus_SNo x0) x1 leaving 4 subgoals.
Let x2 of type ι be given.
Assume H3:
x2 ∈ omega.
Apply H4 with
λ x3 x4 . add_SNo 1 x4 ∈ int.
Apply ordinal_ordsucc_SNo_eq with
x2,
λ x3 x4 . x3 ∈ int leaving 2 subgoals.
Apply nat_p_ordinal with
x2.
Apply omega_nat_p with
x2.
The subproof is completed by applying H3.
Apply Subq_omega_int with
ordsucc x2.
Apply omega_ordsucc with
x2.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H3:
x2 ∈ omega.
Apply H4 with
λ x3 x4 . add_SNo 1 x4 ∈ int.
Apply nat_inv with
x2,
add_SNo 1 (minus_SNo x2) ∈ int leaving 3 subgoals.
Apply omega_nat_p with
x2.
The subproof is completed by applying H3.
Assume H5: x2 = 0.
Apply H5 with
λ x3 x4 . add_SNo 1 (minus_SNo x4) ∈ int.
Apply minus_SNo_0 with
λ x3 x4 . add_SNo 1 x4 ∈ int.
Apply add_SNo_0R with
1,
λ x3 x4 . x4 ∈ int leaving 2 subgoals.
The subproof is completed by applying SNo_1.
Apply Subq_omega_int with
1.
Apply nat_p_omega with
1.
The subproof is completed by applying nat_1.
Apply H5 with
add_SNo 1 (minus_SNo x2) ∈ int.
Let x3 of type ι be given.
Apply H6 with
add_SNo 1 (minus_SNo x2) ∈ int.
Apply H8 with
λ x4 x5 . add_SNo 1 (minus_SNo x5) ∈ int.
Apply ordinal_ordsucc_SNo_eq with
x3,
λ x4 x5 . add_SNo 1 (minus_SNo x5) ∈ int leaving 2 subgoals.
Apply nat_p_ordinal with
x3.
The subproof is completed by applying H7.
Apply minus_add_SNo_distr with
1,
x3,
λ x4 x5 . add_SNo 1 x5 ∈ int leaving 3 subgoals.
The subproof is completed by applying SNo_1.
Apply ordinal_SNo with
x3.
Apply nat_p_ordinal with
x3.
The subproof is completed by applying H7.
Apply add_SNo_minus_SNo_prop2 with
1,
minus_SNo x3,
λ x4 x5 . x5 ∈ int leaving 3 subgoals.
The subproof is completed by applying SNo_1.
Apply SNo_minus_SNo with
x3.
Apply ordinal_SNo with
x3.
Apply nat_p_ordinal with
x3.
The subproof is completed by applying H7.
Apply int_minus_SNo_omega with
x3.
Apply nat_p_omega with
x3.
The subproof is completed by applying H7.
The subproof is completed by applying H2.
Let x3 of type ι → ι → ο be given.
Assume H3: x3 y2 y2.
The subproof is completed by applying H3.