The real parameter test

Let $\mathcal{Z}$ be the direct sum of the two relation-test spaces in (eq:leftscalarcondition) and (eq:rightscalarcondition). Define linear functionals $\alpha ,\beta$ on $\mathcal{Z}$ by evaluating the corresponding joint tests against ${u_{\mu }}^{\rho }$ and $r$, respectively. Then both scalar relation tests are equivalent to