# Definition Explanation for differential cryptanalysis

I am reading Differential Attack from Stinson-Cryptography: Theory and Practice on a toy example of S-box(block Cipher)

I am mainly confused in the following definition

Definition 3.1 : Let $$\pi_S:\{0,1\}^m\rightarrow \{0,1\}^n$$ be a S-box. Consider an (ordered) pair of bitstrings of length $$m$$, say $$(x,x^*)$$. We say that input XOR of the S-box is $$x\oplus x^*$$ and the output XOR is $$\pi_S(x)\oplus\pi_S(x^*)$$.

Now, the point of confusion is how can the output of the S-box be separated as $$\pi_S(x)\oplus\pi_S(x^*)$$ when the input is $$x\oplus x^*$$?

For eg:
We choose the two plaintexts $$x_1$$ and $$x_2$$. Then we have $$x_{12}=x_1\oplus x_2$$
Now, we have round $$1$$ key $$K^1$$, we get $$^1u^1=x_1\oplus K^1$$ and $$^2u^1=x_2\oplus K^1$$
Then, we apply $$\pi_S$$ on the above, we get $$^1v^1=\pi_S(^1u^1)=\pi_S(x_1\oplus K^1)$$ and $$^2v^1=\pi_S(^2v^1)=\pi_S(x_2\oplus K^1)$$
Therefore, $$^1v^1\oplus\enspace ^2v^1=\pi_S(^1u^1)\oplus \pi_S(^2v^1)=\pi_S(x_1\oplus K^1)\oplus \pi_S(x_2\oplus K^1)$$

Now, I don't understand that How, $$^1v^1\oplus\text{ }^2v^1=\pi_S(x_1)\oplus\pi_S(x_2)$$?

Also, I don't know whether I got the definition or not?