Let x0 of type ι be given.
Apply ordinal_ordsucc_SNo_eq with
mul_SNo 2 x0,
λ x1 x2 . odd_nat x1 leaving 2 subgoals.
Apply nat_p_ordinal with
mul_SNo 2 x0.
Apply omega_nat_p with
mul_SNo 2 x0.
Apply mul_SNo_In_omega with
2,
x0 leaving 2 subgoals.
Apply nat_p_omega with
2.
The subproof is completed by applying nat_2.
The subproof is completed by applying H0.
Apply even_nat_odd_nat_S with
mul_SNo 2 x0.
Apply even_nat_2x with
x0.
The subproof is completed by applying H0.