User:Jshflynn/Left Quotient Related to Right Quotient

Theorem

Let $L$ and $K$ be two languages. Then:

$K^{-1}L = (L^R(K^R)^{-1})^R$

Proof

Let $x \in K^{-1}L$.

So: $y \circ x \in L$ for some $y \in K$

So: $x^R \circ z \in L^R$ for some $z \in K^R$

So: $x^R \in L^R(K^R)^{-1}$

So: $x \in (L^R(K^R)^{-1})^{-1}$

$\blacksquare$