User:Caliburn/s/mt/Horizontal Section of Set Difference is Set Difference of Horizontal Sections

Theorem

 * $\paren {E_1 \setminus E_2}^y = E_1^y \setminus E_2^y$

Proof
Note that:


 * $x \in E_1^y \setminus E_2^y$

This is equivalent to:


 * $x \in E_1^y$ and $x \not \in E_2^y$.

That is:


 * $\tuple {x, y} \in E_1$ and $\tuple {x, y} \not \in E_2$

This is the case :


 * $\tuple {x, y} \in E_1 \setminus E_2$

Which is equivalent to:


 * $x \in \paren {E_1 \setminus E_2}^y$

So we have:


 * $x \in E_1^y \setminus E_2^y$ $x \in \paren {E_1 \setminus E_2}^y$.

giving:


 * $\paren {E_1 \setminus E_2}^y = E_1^y \setminus E_2^y$