Why we can add an element on both sides like this? Suppose $A$ is a set equipped with a binary operation $+$.
How I can prove if $a=b$ then $a+c=b+c$? Why we can add an element on both sides like this?

This also confused me when I first started studying math.  A binary operation $+$ on $A$ is a function $+: A \times A \rightarrow A$.  By $a+b$, we mean $+(a,b)$.
The question is why if $b, c \in A$ with $b =c$, then $+(a,b) = +(a,c)$?
The naive way to do this is to just say: it's obvious; two things are "equal" if they are the same.  If $b$ is the same thing as $c$, then $+(a,b)$ is obviously the same thing as $+(a,c)$, because $b$ is the same thing as $c$.  There is no difference in the expressions.  If I add $3$ to the number $2$, that's obviously the same thing as adding $3$ to the number $1+1$, because $1+1$ is the same thing as $2$.
Another way if you want to think more carefully about it: fix $a \in A$.  Then the mapping $x \mapsto +(a,x)$ is a function $f: A \rightarrow A$.  
So this reduces the problem to showing the following: if $f$ is a function from $A$ to $A$, and $b, c \in A$ with $b = c$, then $f(b) = f(c)$.  Can you see why this is tre?
