Given the following two conditions:
$x\succ y$ implies $x+a\succsim y+a$,
$x\prec y$ implies $x+a\precsim y+a$
We want to prove that $\succsim$ is a linear preference.
One of the definition of linear preference is that: $x\succsim y \Leftrightarrow x+a\succsim y+a$
So I am trying to do this:
Since $x\succsim y$ means that $x\succ y$ or $x\sim y$
We already know that $x\succ y$ implies $x+a\succsim y+a$,
all things left is to prove that $x\sim y$ also implies that $x+a\succsim y+a$.