Let x0 of type ι be given.
Apply and3I with
x0 ∈ omega,
x0 ∈ omega,
∃ x1 . and (x1 ∈ omega) (mul_nat x0 x1 = x0) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H0.
Let x1 of type ο be given.
Apply H1 with
1.
Apply andI with
1 ∈ omega,
mul_nat x0 1 = x0 leaving 2 subgoals.
Apply nat_p_omega with
1.
The subproof is completed by applying nat_1.
The subproof is completed by applying mul_nat_1R with x0.