> [!NOTE] Theorem ($O_{2}(\mathbb{R})$ is indeed a group) > The [[Second Orthogonal Group Over The Reals]] $(O_{2}(\mathbb{R}), \circ)$ is a [[Groups|group]] where $\circ$ denotes [[Function Composition|composition]]. *Proof*. $(G0)$ Follows from [[Isometry of The Plane#^1c0497|composition of plane isometries is a plane isometry]] and noting that the composition of two functions which the origin will the origin. $(G1)$ Follows from [[Associativity of Function Composition|associativity of function composition]]. $(G2)$ Cleary the [[Identity Function|identity map]] on $\mathbb{R}^{2}$ is an isometry on $\mathbb{R}^{2}$ that fixes every point, including the origin. $(G3)$ Take $f\in O_{2}(\mathbb{R})$ then [[Isometry of The Plane#^91f6dc|it is either a rotation about the origin or a reflection in the x-axis followed by a rotation about the origin]]. Either way, it has an inverse in the group.