Let x0 of type ι be given.
Let x1 of type ι be given.
Apply nat_ind with
λ x2 . mul_nat (mul_nat x0 x1) x2 = mul_nat x0 (mul_nat x1 x2) leaving 2 subgoals.
Apply mul_nat_0R with
x1,
λ x2 x3 . mul_nat (mul_nat x0 x1) 0 = mul_nat x0 x3.
Apply mul_nat_0R with
x0,
λ x2 x3 . mul_nat (mul_nat x0 x1) 0 = x3.
The subproof is completed by applying mul_nat_0R with
mul_nat x0 x1.
Let x2 of type ι be given.
Apply mul_nat_SR with
mul_nat x0 x1,
x2,
λ x3 x4 . x4 = mul_nat x0 (mul_nat x1 (ordsucc x2)) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply mul_nat_SR with
x1,
x2,
λ x3 x4 . add_nat (mul_nat x0 x1) (mul_nat (mul_nat x0 x1) x2) = mul_nat x0 x4 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply mul_add_nat_distrL with
x0,
x1,
mul_nat x1 x2,
λ x3 x4 . add_nat (mul_nat x0 x1) (mul_nat (mul_nat x0 x1) x2) = x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply mul_nat_p with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.