How prove this definition $a\oplus b=a+b$ Define $\oplus$: if for any real numbers $a,b,c$  there have
$$\left(a\oplus b\right)\oplus c=a+b+c$$
show that
$$a\oplus b=a+b$$

Let $k =0\oplus 0$. 
Since $(0\oplus 0)\oplus c=c$ we see that $k\oplus c=c$ for each $c$, so $k$ is left neutral.
Next we see that $\oplus $ is commutative: 
$$b\oplus c=(k\oplus b)\oplus c=k+b+c=k+c+b=(k\oplus c)\oplus b=c\oplus b$$
Now we see that $k=0$:
$$(a\oplus k)\oplus k=a+2k \Rightarrow a=a+2k \Rightarrow k=0 $$
So $$a+b=a+b+0=(a\oplus b)\oplus 0=a\oplus b$$
