According to Thomas Aquinas, God is a structure $G$ with the elements ‘pater‘, ‘filius‘ and ‘spiritus sanctus‘, in a signature consisting of one asymmetric binary relation (‘relatio opposita‘) $R$, read as ‘relatio originis‘. Aquinas asserts also that the three elements can be uniquely identified the terms of $R^G$. We are to deduce – as Aquinas did – that if the pairs (pater, filius) and (pater, spiritus sanctus) lie in $R^G$, then exactly one of the pairs (filius, spiritus santus) and (spiritus sanctus, filius) lies in $R^G$.
According to Aquinas, $(pater, filius) \in R^G$. Likewise, $(pater, spiritus sanctus)\in R^G$. Then there are only two possibilities left that need to be dealt with in the relation, that of filius and spirtitus sanctus. Either the case is $(filius, spiritus\, santus) \in R^G$ or it is $(spiritus\, sanctus, filius) \in R^G$, or both. Assume it is $(filius, spiritus\, santus)\in R^G$, then $(spiritus\, sanctus, filius) \not \in R^G$, because $R$ is asymmetric. On the other hand, if it is the case that $(spiritus\, sanctus, filius) \in R^G$ then $(filius, spiritus\, santus) \not \in R^G$, since $R^G$ is asymmetric.
$\therefore$ only one of the two lies in $R^G$.