this is a good question! let then by the left cancellation property and so for some let then by associativity we

have: and thus by the left cancellation property and hence the claim is that by the right cancellation property:

thus for any there exists such that thus: now let then and so by the left cancellation property:

thus this proves that so the only thing left is to show that every element of G has an inverse. let then thus there exists such

that so every element of G has a right inverse. particularly for some but then hence and the proof is complete.