Ring/Basic properties/Fact/Proof
Proof
In the noncommutative case, we only proof one half of the statements.
- We have . Subtracting (meaning addition with the negative of ) on both sides gives the claim.
due to part (1). Therefore, is the (uniquely determined) negative of .
- Due to (2), we have , and because of (which holds in every group), we get the claim.
- This follows from the parts proved so far.
- This follows with a double induction.