In the previous exercise (Section 13.14, Exercise #9) we proved the “cab minus bac” formula:

Using this formula prove the following identities:

- .
- .
- if and only if .
- .

*Proof.*Using the cab minus bac formula with in place of , in place of , and in place of we have*Proof.*Applying the cab minus back formula to each of the three terms in the sum we haveSo, putting these together we have

*Proof.*From cab minus bac we haveFurthermore, since , we can apply bac minus cab to get

Therefore,

*Proof.*From a previous exercise (Section 13.14, Exercise #7(d)) we know the identity . In this case we have in place of , in place of and in place of . This gives us