The same theorem forall c != 0, (a+b)/c = a/c + b/c is true before and after. It may be helpful to observe that while using the same symbol '/' in both cases

The same theorem forall c != 0, (a+b)/c = a/c + b/c is true before and after. It may be helpful to observe that while using the same symbol '/' in both cases is confusing, mathematical theorems are not about symbols, but about particular mathematical objects. The old theorem is about the division relation defined for real numbers != 0, the new one is a relation defined for all real numbers, that just happens to coincide.