Proposition 2

In an ordered group G the relations $x \leq y$ and $x + z \leq y + z$ are equivalent.

Indeed one can obtain each from the other by adding $z$ or $(-z)$ to both sides.

This fact can be expressed by saying that, in an ordered group G, the ordering is translation-invariant. In other words a translation is an automorphism of the ordering of an ordered group.