This holds in any ring. Since the reals are a ring, it certainly works.

Proof.

Use use the left-right distributive laws:

a(b+c)=ab+ac

(b+c)a=ba+ca

We know that (xy)+(-xy)=0 by definition.

That is -(xy) is an additive inverse.

We need to show that

xy+(-x)y=0

And hence an additive inverse of xy.

By uniqueness we conclude that -(xy)=(-x)y.

In order to show that, meaning,

xy+(-x)y=0

We use distrivutive laws,

xy+(-x)y=(x+(-x))y=0y=0.

Q.E.D.

(The other proofs are anagolus).

Let us argue by contradiction.c) if x is not equal to 0, then (1/x) is not equal to zero and 1/(1/x) = x

If 1/x=0

Then x(1/x)=x(0)=0

Thus, 1=0

Which is false/