Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Apply H0 with
divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Apply H2 with
(∃ x4 . and (x4 ∈ omega) (mul_nat x0 x4 = x2)) ⟶ divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Apply H5 with
divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Let x4 of type ι be given.
Apply H6 with
divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Apply H1 with
divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Apply H9 with
(∃ x5 . and (x5 ∈ omega) (mul_nat x1 x5 = x3)) ⟶ divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Apply H12 with
divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Let x5 of type ι be given.
Apply H13 with
divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3).
Apply and3I with
mul_SNo x0 x1 ∈ omega,
mul_SNo x2 x3 ∈ omega,
∃ x6 . and (x6 ∈ omega) (mul_nat (mul_SNo x0 x1) x6 = mul_SNo x2 x3) leaving 3 subgoals.
Apply mul_SNo_In_omega with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H10.
Apply mul_SNo_In_omega with
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H11.
Let x6 of type ο be given.
Apply H16 with
mul_SNo x4 x5.
Apply andI with
mul_SNo x4 x5 ∈ omega,
mul_nat (mul_SNo x0 x1) (mul_SNo x4 x5) = mul_SNo x2 x3 leaving 2 subgoals.
Apply mul_SNo_In_omega with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H14.
Apply mul_nat_mul_SNo with
mul_SNo x0 x1,
mul_SNo x4 x5,
λ x7 x8 . x8 = mul_SNo x2 x3 leaving 3 subgoals.
Apply mul_SNo_In_omega with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H10.
Apply mul_SNo_In_omega with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H14.
Apply mul_SNo_com_4_inner_mid with
x0,
x1,
x4,
x5,
λ x7 x8 . x8 = mul_SNo x2 x3 leaving 5 subgoals.