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Closure relationships

It follows that if the gains are antenna based like in equation 1, we have the following closure relationships:
$\displaystyle \widetilde{\varphi}_{ij} + \widetilde{\varphi}_{jk} - \widetilde{\varphi}_{ik}$ $\displaystyle =$ $\displaystyle \varphi_{ij} + \varphi_{jk} - \varphi_{ik}$ (14)
$\displaystyle \frac{\widetilde{a}_{ij}\widetilde{a}_{kl}} {\widetilde{a}_{ik}\widetilde{a}_{jl}}$ $\displaystyle =$ $\displaystyle \frac{a_{ij}a_{kl}}{a_{ik}a_{jl}}$ (15)