Homotopic Paths Implies Homotopic Composition

Theorem
Let $X$ be a topological space.

Let $f_1,f_2,g_1,g_2 : [0,1] \to X$ be paths.

Let $f_1$ be homotopic to $f_2$ and $g_1$ be homotopic to $g_2$.

Then the composed paths $f_1g_1$ and $f_2g_2$ are homotopic.

Proof
Let $F:[0,1]\times[0,1] \to X$ be a homotopy between $f_1$ and $f_2$.

Let $G:[0,1]\times[0,1] \to X$ be a homotopy between $g_1$ and $g_2$.

Define $H:[0,1]\times[0,1] \to X$ by:
 * $\displaystyle H(s,t)= \begin{cases} F(2s, t) & s\in\left[0, \frac12\right] \\

G(2s - 1, t) & s\in\left[\frac12, 1\right] \end{cases}$

By Continuous Mapping on Finite Union of Closed Sets, $H$ is continuous.

By definition of composition of paths, $H$ is a homotopy between $f_1g_1$ and $f_2g_2$.

Also see

 * Definition:Multiplication of Homotopy Classes of Paths