Weierstrass shift-difference identity by pole cancellation ($[\wp (z-a)-\wp (z+a)][\wp (z)-\wp (a){]}^2={\wp }^{\prime }(z){\wp }^{\prime }(a)$)

For $2a\notin \Lambda$, the apparent double poles at $z=\pm a$ on the left are canceled by its squared zero factor. Subtract the right side: the order-three principal parts at lattice points cancel by the even Laurent expansion of $\wp$, leaving at most a simple pole per period cell. An elliptic function has total residue zero in a period cell, so that possible pole is removable. The resulting entire periodic function is constant, and its oddness makes the constant zero. This proves the identity without presupposing the elliptic differential equation.