In this section, we prove the binomial theorem for semirings, srgbinom, which is a generalization of the binomial theorem for complex numbers, binom: is the sum from to of .
Note that the binomial theorem also holds in the non-unital case (that is, in a "rg") and actually, the additive unit is not needed in its proof either. Therefore, it can be proven in even more general cases. An example is the "rg" (resp. "rg without a zero") of integrable nonnegative (resp. positive) functions on .
Special cases of the binomial theorem are csrgbinom (binomial theorem for commutative semirings) and crngbinom (binomial theorem for commutative rings).