User:Caliburn/s/mt/Horizontal Section of Set Difference is Set Difference of Horizontal Sections
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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 if and only if:
- $\tuple {x, y} \in E_1 \setminus E_2$
Which is equivalent to:
- $x \in \paren {E_1 \setminus E_2}^y$
So we have:
- $x \in E_1^y \setminus E_2^y$ if and only if $x \in \paren {E_1 \setminus E_2}^y$.
giving:
- $\paren {E_1 \setminus E_2}^y = E_1^y \setminus E_2^y$