by $(\sigma,\leq)$.

\[proposition-Near-sel-v8:1\] Let $(\sigma,\leq)$ be an oriented simplex and $\Phi,\Psi:\sigma\operatorname{\leadsto}Y$ be mappings into a space $Y$ such that $\Phi{\lhook\joinrel\xrightarrow{\infty}}\Psi$ and $\Psi(v)\subset \Phi(u)$, for every $u,v\in \sigma$ with $u<v$. For every $v\in \sigma$, fix a contraction $H_v:\Phi(v)\times[0,1]\to \Psi(v)$ into some point $p_v\in \Psi(v)$. Then there exists a unique continuous map $h:|\sigma|\to Y$ such that

1.  \[item:Near-sel-v8:1\] $h(v)=p_v$, $v\in \sigma$,

2.  \[item:Near-sel-v8:2\] $h\big(tv+(1-t)z\big)=H_v(h(z),t)$, $z\in |\tau|$ and $t\in [0,1]$, whenever $\tau\subset \sigma$ is a face and $v\in \sigma$ with $v< u$ for every $u\in \tau$.

Suppose that $\sigma=\{v_0,\dots, v_{n+1}\}$, where $v_0<v_1<\dots <v_{n+1}$, and consider the maximal chain $\tau_{n+1}\subset \tau_n\subset\dots \subset \tau_0=\sigma$ of faces of $\sigma$ defined by $\tau_{n+1}=\{v_{n+1}\}$ and $\tau_k=\tau_{k+1}\cup\{v_k\}$, $k\leq n$. We will construct the required map $h:|\sigma|\to Y$ inductively on the faces $|\tau_k|$, $k\leq n+1$, of this chain. To this end, set $S_k=\Phi(v_k)$, $T_k=\Psi(v_k)$, $H_k=H_{v_k}$ and $p_k=p_{v_k}$, $k\leq n+1$. Next, define $h_{n+1}:|\tau_{n+1}|\to T_{n+1}$ by $h_{n+1}(v_{n+1})=p_{n+1}$. Since $|\tau_n|=|\tau_{n+1}|*v_n$ and $T_{n+1}=\Psi(v_{n+1})\subset \Phi(v_n)=S_n$, by Proposition \[proposition-Near-sel-v7:2\], $h_{n+1}$ has a continuous extension $h_n:|\tau_n|\to T_n$ such that $$\begin{aligned}
    h_n\big(tv_n+(1-t)z\big)&=H_n(h_{n+1}(z),t)\\
                            &=H_n(h_n(z),t),\quad\text{$z\in
                              |\tau_{n+1}|$ and $t\in [0,1]$.} 
  \end{aligned}$$ It is also evident that $h_n(v_n)=H_n(p_{n+1},1)=p_n$. The construction can be carried on by a finite induction to get a continuous map $h=h_0:|\tau_0|=|\sigma|\to T_0$ satisfying \[item:Near-sel-v8:1\] and such that for every 