$h_2(k)=q_2$. Define $$h(i){\equiv}\left\{
    \begin{array}{ll}
     h_2(1)*h_1(i) & i\in\{1,\ldots,n\} \\
     h_2(i-n)*h_1(n) & i\in\{n+1,\ldots,n+k\}      
    \end{array}\right.$$ Then $h:\{1,\ldots,n+k\}\longrightarrow \mathrm{P}(a_0,a_2)$ is a homotopy of paths such that $h(1)= p_2*p_1$ and $h(n+k)=q_2*q_1$, completing the proof.

The *reverse* of a 1-simplex $b$, is the 1-simplex ${\overline{b}}$ defined as $$\label{Ab:6}
\partial_0{\overline{b}} = \partial_1b, \ \ \  
\partial_1{\overline{b}} = \partial_0b, \ \ \ 
|{\overline{b}}| = |b|.$$ So, the *reverse* of a path $p=\{b_n,\ldots,b_1\}\in\mathrm{P}(a_0,a_1)$ is the path ${\overline{p}}\in\mathrm{P}(a_1,a_0)$ defined as ${\overline{p}}{\equiv}\{ {\overline{b_1}},\ldots,{\overline{b_n}}\}$. It is clear that ${\overline{{\overline{p}}}}=p$. Furthermore

\[Ab:8\] $p\sim q$ $ \ \  \Rightarrow \ \ $ ${\overline{p}}\sim{\overline{q}}$.

The reverse of a 2-simplex $c$ is the 2-simplex ${\overline{c}}$ defined as $$\partial_1{\overline{c}}={\overline{\partial_1c}}, \ \ 
\partial_0{\overline{c}}={\overline{\partial_2c}}, \ \ 
\partial_2{\overline{c}}={\overline{\partial_0c}}, \ \ 
|{\overline{c}}|=|c|.$$ Note that, if $c\in{\mathsf{A}}(b)$, then ${\overline{c}}\in{\mathsf{A}}({\overline{b}})$; if $c\in{\mathsf{C}}(b_2,b_1)$, then ${\overline{c}}\in{\mathsf{C}}\big( {\overline{b_1}},{\overline{b_2}}\big)$. So, let $h:\{1,\ldots,n\}\longrightarrow \mathrm{P}(a_0,a_1)$ be a homotopy of paths. Then maps ${\overline{h}}:\{1,\ldots,n\}\longrightarrow \mathrm{P}(a_1,a_0)$ defined as ${\overline{h}}(i){\equiv}{\overline{h(i)}}$ for any $i$ is a homotopy of paths, completing the proof.

A 1-simplex $b$ is said to be *degenerate to* a 0-simplex $a_0$ whenever $$\label{Ab:9}
 \partial_0 b =a_0 =\partial_1b, \ \ \ \  a_0=|b|$$ We will denote by $b(a_0)$ the 1-simplex degenerate to $a_0$.

\[Ab:10\] The following assertions hold:\
(a) $p*b(\partial_1p)\sim p \sim b(\partial_0p)*p$;\
(b) $p*{\overline{p}}\sim b(\partial_0p)$ and ${\overline{p}}*p\sim b(\partial_1p)$.

By Lemma 