While studying real analysis, I got confused on the following issue.
Suppose we construct real numbers as equivalence classes of cauchy sequences. Let $x = (a_n)$ and $y= (b_n)$ be two cauchy sequences, representing real numbers $x$ and $y$.
Addition operation $x+y$ is defined as $x+y = (a_n + b_n)$.
To check if this operation is well defined, we substitute $x = (a_n)$ with some real number $x' = (c_n)$ and verify that $x+y = x'+y$. We also repeat it for $y$. i.e. we verify that $x+y = x+y'$.
Instead of checking that $x+y = x+y'$ and $x'+y = x+y$ seperately, would it suffice to check that $x+y = x' + y'$ in a single operation in order to show that addition is well defined for real numbers. Would it hurt to checking well definedness? Can any one explain me the logic behind ?