Let x0 of type ι be given.
Let x1 of type ι be given.
Apply nat_ind with
λ x2 . CRing_with_id_omega_exp x0 x1 x2 ∈ field0 x0 leaving 2 subgoals.
Apply CRing_with_id_omega_exp_0 with
x0,
x1,
λ x2 x3 . x3 ∈ field0 x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply CRing_with_id_one_In with
x0.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Apply CRing_with_id_omega_exp_S with
x0,
x1,
x2,
λ x3 x4 . x4 ∈ field0 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply nat_p_omega with
x2.
The subproof is completed by applying H2.
Apply CRing_with_id_mult_clos with
x0,
x1,
CRing_with_id_omega_exp x0 x1 x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Apply L2 with
x2.
Apply omega_nat_p with
x2.
The subproof is completed by applying H3.