Definition

Let $f$ be a path from $x_0$ to $x_1$ and $g$ a path from $x_1$ to $x_2$. Then their product

$f *g: I \to X$,

$(f*g)(s) = \begin{cases} f(2s)\quad 0 \leq s \frac{1}{2}\\ g(2s-1)\quad \frac{1}{2} \leq s \leq 1 \end{cases}$

Use Pasting lemma to show it is a path

(remember: defined only if $f(1) = g(0)$

vid