If statements are formulated entirely within a first-order language for the reals, then they are true of the standard reals if and only if they are true for the hyperreals

In fact, Robinson’s hyperreals satisfy a “transfer principle”—if statements are formulated entirely within a first-order language for the reals, then they are true of the standard reals if and only if they are true for the hyperreals. So any proof of such a theorem in one system can be transferred to the other. This sometimes greatly simplifies calculations and proofs of theorems.