Definition:Ordered Sum/Caution

Definition
Note the way the definition of an ordered sum has been worded.

Suppose $a, b \in S \cap T$.

Suppose:
 * $a \preceq_1 b$ (through dint of $a, b \in S$
 * $b \preceq_2 a$ (through dint of $a, b \in T$.

Then because $a, b \in S$, we have that $a \prec b$.

But, also because $a, b \in S$, we do not consider the fact that $a, b \in T$ and so the relation $b \preceq_2 a$ is ignored.