The second term is zero, which can be proved using the same approach as for the left bound.

The first two conditions are satisfied simply by the definition of g, while the third condition can be proved using polynomial long division.

This identity can be easily proved using the matrix multiplication representation.

Boole's inequality may be proved using the method of induction.

It was the first major theorem to be proved using a computer.

This can be proved using similar methods used by Newman for his proof of the prime number theorem.

The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.

This may be proved using an ordering on the noncommutative monomials.

This can be proved using the following observation.

It can itself be proved using mathematical induction, as shown below.