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We solve a conditional functional equation of the form $$x{\perp }^{\rho }y\Rightarrow f(x+y)=f\left(x\right)+f\left(y\right),$$ where $f$ is a mapping from a real normed linear space $(X,\parallel ·\parallel )$ with $\text{dim}X\ge 2$ into an abelian group $(G,+)$ and ${\perp }^{\rho }$ is a given orthogonality relation associated to the norm.