$f(x_1,y_1)=f(x_2,y_2)$ $\implies$ $||f(x_1,y_1)||=||f(x_2,y_2)||$

Question

How is the following shown:

$$f(x_1,y_1)=f(x_2,y_2) \implies ||f(x_1,y_1)||=||f(x_2,y_2)||$$

?

That is, that if two elements are equivalent, then their norms are equivalent.