= Solution
Work on a common stochastic basis on which $B,B'$ are independent <Brownian motions> and the solutions are adapted; independent weak solutions can be put on the product stochastic basis. Independence gives $[B,B']=0$. Put $S=Z+Z'\geq0$ and define the predictable weights
$$
u_t=\begin{cases}\sqrt{Z_t/S_t},&S_t>0,\\1,&S_t=0,\end{cases}\qquad
v_t=\begin{cases}\sqrt{Z'_t/S_t},&S_t>0,\\0,&S_t=0.\end{cases}
$$
They are bounded and satisfy $u_t^2+v_t^2=1$ everywhere. Define
$$
\boxed{\beta_t=\int_0^tu_s\,dB_s+\int_0^tv_s\,dB'_s.}
$$
This is a zero-start <continuous local martingale>, and its <quadratic variation> is $[\beta]_t=\int_0^t(u_s^2+v_s^2)ds=t$. By the <Lévy characterization of Brownian motion>, $\beta$ is a <Brownian motion> in the common filtration.
On $\{S_t>0\}$, $\sqrt{S_t}u_t=\sqrt{Z_t}$ and $\sqrt{S_t}v_t=\sqrt{Z'_t}$. On $\{S_t=0\}$, nonnegativity forces both $Z_t,Z'_t$ to be zero, so the same identities hold. <Associativity of stochastic integration> therefore gives
$$
S_t=z+z'+\int_0^t\sqrt{S_s}\,d\beta_s.
$$
This proves the <addition law for square-root branching diffusions>: \b[$Z+Z'$ is a weak solution with initial value $z+z'$]. Specifying the weights at zero is necessary to keep $[\beta]_t=t$ after extinction; simply setting both weights to zero would fail to produce a Brownian driver.
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