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

We consider

${\displaystyle \varphi \circ {\begin{pmatrix}1&0\\0&-1\end{pmatrix}},}$

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

${\displaystyle {}\varphi \circ {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}={\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}\,.}$

Therefore,

${\displaystyle {}\varphi ={\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}\circ {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}={\begin{pmatrix}\cos \alpha &\sin \alpha \\\sin \alpha &-\cos \alpha \end{pmatrix}}\,.}$