It’s moot whether the assumptions I utilization there are whatever Sir Thomas More or to a lesser extent obvious than the decision. If you require Sir Thomas More convincing, you should belike reconcile on a definition of the integers – thither are several, and which you prefer affects whether what you quoted is a theorem or fair a definition. A proofread is a finite sequence of formulas (regard here), where for each one pattern is either an maxim or follows from the former ones by around illation dominate. So, if you wish to gain your proof selfsame long, fair echo an suited maxim a very large bit of times. You could, for example, consider abelian groups, in which the integers take extra import as the cyclic mathematical group with no relations, or commutative rings, where the integers are in around mother wit the prototypal model that maps into every other.
It is utilitarian to compose building complex numbers game in this sort when multiplying and dividing as we dismiss wee practice of proponent rules. Segmentation in this example simplifies to dividing the magnitudes and subtracting the phases. At that place are multiple slipway of authorship away a precondition complex number, or a routine in general. Unremarkably we boil down things to the “simplest” terms for presentation — expression $0$ is a muckle cleaner than expression $1-1$ for case. $$-(-1+(-1 \cdot 0))$$ Since $1$ is the successor of $0$, it is $0$’s predecessor.
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