*or, what number theorists do for a good time.*
This

proof uses only the fundamental

axioms of the

integers.

0 = 0 {reflexive property of equality}

0 + 0 = 0 {additive identity}

A(0 + 0) = A(0) {multiplication is well defined}

A(0) + A(0) = A(0) {distribution of * over +}

A(0) + A(0) + -A(0) = A(0) + -A(0) {addition is well defined}

A(0) + (A + -A)(0) = (A + -A)(0) {distribution of * over +}

A(0) + (0)(0) = (0)(0) {additive inverses}

A(0) = 0 {additive identity}

Q.E.D.