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I have proved step 1 to 6. However, I am stopped by step 7. Here's the quote to Baby Rudin

We complete the definition of multiplication by setting $\alpha 0^*=0^*=0^*\alpha$ and by setting

$$ \alpha\beta= \begin{cases} (-\alpha)(-\beta),\text{ if }\alpha<0^*,\beta <0^*\\ -[(-\alpha)\beta],\text{ if }\alpha < 0^*,\beta > 0^*\\ -[\alpha(-\beta)],\text{ if }\alpha > 0^*,\beta < 0^* \end{cases} $$

My question is, can these be proved?

I think no. So for example, let's take the first case as a demonstration.

Addition axiom holds for positive and negative reals for now, which is basically the entire real number set. So it suffices to prove $\alpha\beta=-[(-\alpha)\beta]$ if $-\alpha\beta+\alpha\beta=-\alpha\beta+-[(-\alpha)\beta]$ and $\alpha <0^*,\beta >0^*$.

To me, I find this extremely easy if distributive axiom holds in the set of reals, because I can just factor out the common factor negative alpha and apply addition inverse as well as product includes 0 will be 0. However, both distributive axiom and the product includes 0 will be 0 warrants the distributive axiom, which is yet defined in the set of reals for now. In fact, it will be proved afterwards according to baby rudin.

For now, I can't prove these settings. Let say I take it for granted and prove multiplication axioms hold in the set of reals. But how about these settings? How can I prove them?

The above were my efforts in proving according to established lemmas in the book. the following are some links that I have researched. Since it is possible for me to have overlooked some post which may possibly answer my question, for those who have a duplicate post please attach after checking duplicates.

Question regarding step 8 in appendix of chapter 1 , baby rudin

Step 6 Real construction from Rational Baby Rudin

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Ok, I think I kind of understand it.

Multiplication in the negative cuts are not defined now. So Rudin used established constructs, namely, the positive reals, to define multiplication in negative cuts.

The validity is predicated upon the lack of definition of negative multiplication.

In other words, this is a definition. Just that somehow I thought it is related to any existing theorems.

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