Affine Group of One Dimension is Group

Theorem
The $1$-dimensional affine group on $\R$ $\operatorname{Af}_1 \left({\R}\right)$ is a group.

Proof
Taking the group axioms in turn:

G0: Closure
Let $f_{ab}, f_{cd} \in \operatorname{Af}_1 \left({\R}\right)$.

Then:

By the field axioms, $a c \in \R_{\ne 0}$ and $a d + b \in \R$.

Thus $f_{ab} \circ f_{cd} \in \operatorname{Af}_1 \left({\R}\right)$ and so $\operatorname{Af}_1 \left({\R}\right)$ is closed.

G1: Associativity
From Composition of Mappings is Associative, it follows directly that $\circ$ is associative on $\operatorname{Af}_1 \left({\R}\right)$.

G2: Identity
By Identity of Affine Group of One Dimension, $\operatorname{Af}_1 \left({\R}\right)$ has $f_{1, 0}$ as an identity element.

G3: Inverses
By Inverse in Affine Group of One Dimension, every element $f_{a b}$ of $\operatorname{Af}_1 \left({\R}\right)$ has an inverse $f_{c d}$ where $c = \dfrac 1 a$ and $d = \dfrac {-b} a$.

All the group axioms are thus seen to be fulfilled, and so $\operatorname{Af}_1 \left({\R}\right)$ is a group.