Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply and3E with
x0 ∈ omega,
x1 ∈ omega,
∃ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1),
divides_nat x0 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply H4 with
divides_nat x0 x2.
Let x3 of type ι be given.
Apply H5 with
divides_nat x0 x2.
Apply and3E with
x1 ∈ omega,
x2 ∈ omega,
∃ x4 . and (x4 ∈ omega) (mul_nat x1 x4 = x2),
divides_nat x0 x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H10 with
divides_nat x0 x2.
Let x4 of type ι be given.
Apply H11 with
divides_nat x0 x2.
Apply and3I with
x0 ∈ omega,
x2 ∈ omega,
∃ x5 . and (x5 ∈ omega) (mul_nat x0 x5 = x2) leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H9.
Let x5 of type ο be given.
Apply H14 with
mul_nat x3 x4.
Apply andI with
mul_nat x3 x4 ∈ omega,
mul_nat x0 (mul_nat x3 x4) = x2 leaving 2 subgoals.
Apply nat_p_omega with
mul_nat x3 x4.
Apply mul_nat_p with
x3,
x4 leaving 2 subgoals.
Apply omega_nat_p with
x3.
The subproof is completed by applying H6.
Apply omega_nat_p with
x4.
The subproof is completed by applying H12.
Apply mul_nat_asso with
x0,
x3,
x4,
λ x6 x7 . x6 = x2 leaving 4 subgoals.
Apply omega_nat_p with
x0.
The subproof is completed by applying H2.
Apply omega_nat_p with
x3.
The subproof is completed by applying H6.
Apply omega_nat_p with
x4.
The subproof is completed by applying H12.
Apply H7 with
λ x6 x7 . mul_nat x7 x4 = x2.
The subproof is completed by applying H13.