Euclidean plane/Improper isometry/Axis reflection/Fact/Proof

Proof

We consider

$\phi \circ \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),$

which is by the multiplication theorem for the determinant a proper isometry. Because of fact, there exists a uniquely determined angle $\alpha \in [0,\pi [$ such that

$\phi \circ \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right).$

Therefore,

$\phi =\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)\circ \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ \sin \alpha & -\cos \alpha \end{array}\right).$

To fact