# Proving that operations give equal results given equal inputs

I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as assuming $1 \neq 0$ (although I get that now).

But I got stuck when I wanted to conclude stuff $a \cdot c = b \cdot c$ from $a = b$ where $\cdot$ means addition, multiplication or w/e. There was no property dealing with these cases!

Someone on IRC told me that a property stating that operations give equal results given equal inputs (note: the order of the inputs is preserved; we aren't dealing with commutativity here) was implicitly assumed in the book.

Also, last year in school, we learnt about Euclid's axioms which explicitly stated stuff like this (and in a rather wordy and arbitrary way too, it seemed; unlike the rest of this group and field stuff).

My questions are:

1. What is this property actually called?
2. Does it apply to every single operation definable (except stuff like RNGs)?
3. Why wasn't this mentioned along-side other 'basic' facts in the book?
• I don't quite understand what the problem is. If $a=b$ then you may interchange $a$ and $b$ since they are the same. They are just different letters representing the same thing. Thus, $a\cdot c=b\cdot c$. QED. May 5, 2014 at 9:25
• @IttayWeiss On reflection, what you say makes sense. If $a+b=c$, then indeed $c$ is just another name for the abstract number also called $a+b$ (like C compiler text macros). I think one reason I got confused was due to Euclid's axioms (see my edit). May 5, 2014 at 14:15
• Also, just as when the properties mention $+$, they really mean any binary operation that satisfies those properties (like OR wth only 0 and 1); couldn't there be like different meanings of $=$, like in modulo arithmetic or w/e? I don't know if you'd view 1 and 5 as the same thing in mod 4 arithmetic; more like equivalent integers or something? May 5, 2014 at 14:17
• On a semi-related note: this BetterExplained article on equality helped... May 5, 2014 at 14:20

$$x = y → φ [z := x] → φ [z := y]$$