I am currently reading through Rudin's Principles of Mathematical Analysis and I am learning about fields and their properties. Note that this is the initial chapter - I am just starting off.  I was wondering which field property enables us to multiply on both sides of an equation and still preserve equality.  There is a very clear proposition stated in the book that gives me this for inequalities: $$ \text{If} \ \ x &gt; 0 \ \ \text{and} \ \ y &lt; z \ \ \text{then} \ \ xy&lt;xz. $$  However, the only proposition that seems useful for this in the case of equalities, is stated as an implication and not an equivalence: $$ \text{If} \ x\not= 0 \ \ \text{and} \ \ xy=xz \ \ \text{then} \ \ y=z. $$  Any help would be much appreciated.
It seems that given a statement $a = b$, that $a + c = b + c$ is assumed also to be true.  Why isn't this an axiom of arithmetic, like the commutative law or associative law?  Or is it a consequence of some other axiom of arithmetic?  Thanks!  Edit: I understand the intuitive meaning of equality. Answers that stated that $a = b$ means they are the same number or object make sense but what I'm asking is if there is an explicit law of replacement that allows us to make this intuitive truth a valid mathematical deduction. For example is there an axiom of Peano's Axioms or some other axiomatic system that allows for adding or multiplying both sides of an equation by the same number?   In all the texts I've come across I've never seen an axiom that states if $a = b$ then $a + c = b + c$. I have however seen if $a &lt; b$ then $a + c &lt; b + c$. In my view $&lt;$ and $=$ are similar so the absence of a definition for equality is strange.
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