Equation (6): Weierstrass elliptic addition identity

For every pair $L=(\omega_1,\omega_2)$ of complex numbers linearly independent over $\mathbb R$, let $\Lambda=\{m\omega_1+n\omega_2:m,n\in\mathbb Z\}$ and define, for every $w\in\mathbb C$, $P(w)=\sum_{\ell\in\Lambda}'\bigl(1/(w-\ell)^2-1/\ell^2\bigr)$ and $D(w)=-\sum_{\ell\in\Lambda}'2/(w-\ell)^3$, where the primed sums are unordered infinite sums, assigned the value $0$ if the corresponding family is not summable, and complex division is total with $a/0=0$ for every $a\in\mathbb C$; in particular, the $\ell=0$ term of $P(w)$ is $1/w^2$. For all $z,v\in\mathbb C$ satisfying $z\notin\Lambda$, $v\notin\Lambda$, and $z+v\notin\Lambda$, the assertion is

The three lattice exclusions also exclude $z=0$, $v=0$, and $z+v=0$ and ensure that $w-\ell\ne0$ for each $w\in\{z,v,z+v\}$ and $\ell\in\Lambda$. No inequality between $z$ and $v$, or between $P(z)$ and $P(v)$, is assumed: the case $z=v$ is included whenever $z,2z\notin\Lambda$, and whenever $P(z)=P(v)$ under the stated hypotheses the asserted equality reduces to $\bigl(D(v)-D(z)\bigr)^2=0$.