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.

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