Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply mul_nat_com with
add_nat x0 x1,
x2,
λ x3 x4 . x4 = add_nat (mul_nat x0 x2) (mul_nat x1 x2) leaving 3 subgoals.
Apply add_nat_p with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply mul_nat_com with
x0,
x2,
λ x3 x4 . mul_nat x2 (add_nat x0 x1) = add_nat x4 (mul_nat x1 x2) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply mul_nat_com with
x1,
x2,
λ x3 x4 . mul_nat x2 (add_nat x0 x1) = add_nat (mul_nat x2 x0) x4 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply mul_add_nat_distrL with
x2,
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0.
The subproof is completed by applying H1.