Inverse in Affine Group of One Dimension

Theorem
Let:
 * $a, c \in \R_{\ne 0} \land b, d \in \R$

Let:
 * $\map {\operatorname {Af}_1} \R$

denote the $1$-dimensional affine group on $\R$.

Let:
 * $f_{a b} \in \map {\operatorname {Af}_1} \R$

Let:
 * $c = \dfrac 1 a \land d = \dfrac {-b} a$

Then:
 * $f_{c d} \in \map {\operatorname {Af}_1} \R$

is the inverse of $f_{a b}$.

Proof
Let:
 * $a, c \in \R_{\ne 0} \land b, d \in \R$

Let:
 * $f_{a b} \in \map {\operatorname {Af}_1} \R$

As:
 * $a \in \R_{\ne 0}$

by definition of:
 * $\map {\operatorname {Af}_1} \R$

it follows that:
 * $\dfrac 1 a \in \R_{\ne 0} \land \dfrac {-b} a \in \R$

Let:
 * $c = \dfrac 1 a \land d = \dfrac {-b} a$

Then:

Similarly:

Hence the result.