Let x0 of type ι be given.
Apply and3I with
x0 ∈ omega,
0 ∈ omega,
∃ x1 . and (x1 ∈ omega) (mul_nat x0 x1 = 0) leaving 3 subgoals.
Apply nat_p_omega with
x0.
The subproof is completed by applying H0.
Apply nat_p_omega with
0.
The subproof is completed by applying nat_0.
Let x1 of type ο be given.
Apply H1 with
0.
Apply andI with
0 ∈ omega,
mul_nat x0 0 = 0 leaving 2 subgoals.
Apply nat_p_omega with
0.
The subproof is completed by applying nat_0.
The subproof is completed by applying mul_nat_0R with x0.