(related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
It is not possible that both cases
are simultaneously true, because it would contradict the cancellation law for adding natural numbers, according to which \(x=y\) implies \(x+u=y+u\) for all natural numbers \(u\).
For the same reason, both cases
cannot be simultaneously true, because the equality \(x=y\) is symmetric and implies \(y=x\).
The cases
can also not be simultaneously true. Otherwise, it would follow from the associativity of adding natural numbers that
\[\begin{array}{rcll} x&=&y+u&\text{assumption case 1}\\ &=&(x+v)+u&\text{assumption case 2}\\ &=&x+(v+u),&\text{associativity of adding natural numbers} \end{array}\]
which is a contradiction.