Ror Riemann's zeta function if one proves a relation in the domain $\operatorname{Re}(s)>1$ will this be sufficient proof that it is satisfied in all the z domain
for example if one proves $\zeta(s)=\overline{\zeta\left(\overline s\right)}$ from the original function defined in $\operatorname{Re}(s)>1$ does it require another proof for $\operatorname{Re}(s)<1$ or can we say that it is satisfied for all z domain and why?
The general question is too broad, but it is easy to answer in the case of the specific example that you mentioned. Since $\zeta(s)=\overline{\zeta\left(\overline s\right)}$ is an equality between analytic functions, then, by the identity theorem and because the domain of the $\zeta$ function is connected, if this identity holds when $\operatorname{Re}s>1$, it holds everywhere.