Let x0 of type ι be given.
Let x1 of type ι be given.
Apply and3I with
x0 ∈ omega,
mul_nat x0 x1 ∈ omega,
∃ x2 . and (x2 ∈ omega) (mul_nat x0 x2 = mul_nat x0 x1) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply nat_p_omega with
mul_nat x0 x1.
Apply mul_nat_p with
x0,
x1 leaving 2 subgoals.
Apply omega_nat_p with
x0.
The subproof is completed by applying H0.
Apply omega_nat_p with
x1.
The subproof is completed by applying H1.
Let x2 of type ο be given.
Apply H2 with
x1.
Apply andI with
x1 ∈ omega,
mul_nat x0 x1 = mul_nat x0 x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι → ι → ο be given.
Assume H3: x4 y3 y3.
The subproof is completed by applying H3.