I think I understand your point: we don't define existence per se. But here we only need to define the existence of the value of a property. And we define it by

I think I understand your point: we don't define existence per se. But here we only need to define the existence of the value of a property. And we define it by saying that if the we can predict the value before measuring it, then the value "exists". Take it as an axiom. When I said "the very value of the entangled properties does not exist locally" in my first post I meant that there is no way to predict the value (or of consistently assigning a value to the property) before performing a measurement.