Field/Basic properties/Fact/Proof2
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Proof
- We have . Subtracting (meaning addition with the negative of ) on both sides gives the claim.
- See exercise.
- See exercise.
- See exercise.
- We prove this by contradiction, so we assume that
and
are both not . Then there exist inverse elements
and
and hence
.
On the other hand, we have
by the premise and so the annulation rule gives
hence , which contradicts the field properties.
- This follows with a double induction, see exercise.