Now one of the axioms of an ordered field is that ; and it is well established that (see here for instance.)
We can now prove the first of the required statements by contradiction,as follows:
First consider the case where and assume that . Then:
using the result above.
But . Contradiction. Therefore .
Then prove it the other way around, and consider the case where , and assume that .
, again using the same result.
But . Contradiction. Therefore
We have now proved the implication in both directions. Hence
To prove the second of your two statements, establish first that . Hint: for some . Then use the distributive law on .
Then use a contradiction method as I did for the first part, to show that . Hence, using the result above that .
Finally, prove the result in the other direction to show that .