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The persons are distinguished on account of their relations, specifically because the relations imply an opposition.
It is therefore better to say that the persons or hypostases are distinguished rather by relations than by origin. For, although in both ways they are distinguished, nevertheless in our mode of understanding they are distinguished chiefly and firstly by relations; whence this name "Father" signifies not only a property, but also the hypostasis; whereas this term "Begetter" or "Begetting" signifies property only; forasmuch as this name "Father" signifies the relation which is distinctive and constitutive of the hypostasis; and this term "Begetter" or "Begotten" signifies the origin which is not distinctive and constitutive of the hypostasis. Summa, I Q40 A2
Hence, the Father is not the Son (person) and paternity is not filiation (properties) because those properties are opposed yet Father, Son, paternity and filiation are the Divine Essence because everything attributed to God is the Divine essence except a relation (because God is simple).