Prove Corollary (1.19).
Corollary (1.19) is
Let $\gcd (a, b) = 1$ (relatively prime) and $x_0$, $y_0$ be particular solutions of the equation $ax + by= c$.
Then all the other solutions of this equation are given by $x = x_0 + bt$ and $y= y_0− at$ where $t$ is any integer.
This corollary is a special case of Proposition (1.18)
Let $\gcd (a, b) = g$. If $g \mid c$ and $x_0$, $y_0$ are particular solutions of the equation $ax + by= c$, then all the other solutions of this equation are given by
$$x = x_0 + (\frac{b}{g})t \quad \text{and} \quad y= y_0− (\frac{a}{g})t$$
where $t$ is any integer.
We are given $g = \gcd(a,b)=1$, so Proposition 1.18 simply reduces to Corollary 1.19.