Ring/Basic properties/Fact/Proof

Proof

In the noncommutative case, we only proof one half of the statements.

  1. We have . Subtracting (meaning addition with the negative of ) on both sides gives the claim.
  2. due to part (1). Therefore, is the (uniquely determined) negative of .

  3. Due to (2), we have , and because of (which holds in every group), we get the claim.
  4. This follows from the parts proved so far.
  5. This follows with a double induction.