TITLE: Correct notational use of $:=$
QUESTION [5 upvotes]: Suppose I use the notation $:=$ to define some symbol the first time it is introduced, e.g. $A:=\Im(f+f^2)$. If later on I make use/remind the reader of the symbol definition, e.g. $A=\Im(f+f^2)=\ldots$, should I use the notation "$:=$", or simply write "$=$" as I did above?

REPLY [3 votes]: This is more like a long comment.
We can view $:=$ as an imperative (its a "command"; it does something). In this case, it probably should not be repeated. We write $x := \tau$ once, and once only, since this has the effect of defining $x$. After this, we always write: "Since $x = \tau...$" 
We can view $:=$ as a strengthening of equality ($x := \tau$ means its true that $x$ is defined to equal $\tau$.) In this case, we can actually reason about statements involving $:=.$ For example, supposing $x:=y$, we can deduce $x=y$. But assuming $x=y$, we cannot deduce $x:=y$. Under this interpretation, repeating $:=$ multiple times is perfectly reasonable.