Homotopy between composites of homotopic loops

Statement

Suppose $X$ is a topological space, $x_0$ is a point in $X$, and $f_1,g_1,f_2,g_2$ are loops based at $x_0$ with the property that $f_1$ is homotopic to $g_1$ (as a loop based at $x_0$) and $f_2$ is homotopic to $g_2$ (again, as a loop based at $x_0$). Then, $f_1*f_2$ is homotopic to $g_1*g_2$.