Let $\Delta_q$ be the standard $q$-simplex. The function $F_q^s : \Delta_{q-1} \to \Delta_q$ is the $s$-th face of $\Delta_q$, defined as the correstriction to $\Delta_q$ of $(e_0,\dots,e_{s-1},\hat{e_s},e_{s+1},\dots,e_q)$, the map $\Delta_{q-1} \to \mathbb{R}^q$ that sends $e_i$ to itself for $i < s$ and $e_i$ to $e_{i+1}$ if $i \geq s$.
I'm stuck on trying to prove the following elementary result,
Lemma. Let $0 \leq j < i \leq q$. Then $F_q^{i}F_{q-1}^{j} = F_q^{j}F_{q-1}^{i-1}$.
I apologize for the simplicity of the question, but I cannot seem to pinpoint where am I making the mistake, so any help is greatly appreciated.
By a direct calculation,
$$ \begin{align} F_q^{i}F_{q-1}^{j}(e_l) &= \cases{F_q^{i}(e_l) \ \text{ if $l < j$}\\ F_q^{i}(e_{l+1}) \text{ if $l \geq j$}} \\&= \cases{ e_l \quad \text{ if $l < j, l< i$}\\ e_{l+1} \ \text{ if $l < j, l \geq i$}\\ e_{l+1} \ \text{ if $l \geq j, l < i$}\\ e_{l+2} \ \text{ if $l \geq j, l \geq i$}} \\&= \cases{ e_l \quad \text{ if $l < j$}\\ e_{l+1} \ \text{ if $l \geq j, l < i$}\\ e_{l+2} \ \text{ if $l \geq i$}} \end{align} $$
and
$$ \begin{align} F_q^{j}F_{q-1}^{i-1}(e_l) &= \cases{F_q^{j}(e_l) \ \text{ if $l < i-1$}\\ F_q^{j}(e_{l+1}) \text{ if $l \geq i-1$}} \\ &= \cases{ e_l \quad \text{ if $l < i-1, l< j$}\\ e_{l+1} \ \text{ if $l < i-1, l \geq j$}\\ e_{l+1} \ \text{ if $l \geq i-1, l < j$}\\ e_{l+2} \ \text{ if $l \geq i-1, l \geq j$}} \\ &= \cases{ e_l \quad \text{ if $l< j$}\\ e_{l+1} \ \text{ if $l < i-1, l \geq j$}\\ e_{l+2} \ \text{ if $l \geq i-1$}} \end{align} $$
These don't seem to coincide. Where have I gone wrong?
You did not correctly calculate $F^s_q(e_{l+1})$. You have $$ \begin{align} F_q^{i}F_{q-1}^{j}(e_l) &= \cases{F_q^{i}(e_l) \ \text{ if $l < j$}\\ F_q^{i}(e_{l+1}) \text{ if $l \geq j$}} \\&= \cases{ e_l \quad \text{ if $l < j, l< i$}\\ e_{l+1} \ \text{ if $l < j, l \geq i$}\\ e_{l+1} \ \text{ if $l \geq j, l+1 < i$}\\ e_{l+2} \ \text{ if $l \geq j, l+1 \geq i$}} \\&= \cases{ e_l \quad \text{ if $l < j$}\\ e_{l+1} \ \text{ if $l \geq j, l < i-1$}\\ e_{l+2} \ \text{ if $l \geq i-1$}} \end{align} $$
and
$$ \begin{align} F_q^{j}F_{q-1}^{i-1}(e_l) &= \cases{F_q^{j}(e_l) \ \text{ if $l < i-1$}\\ F_q^{j}(e_{l+1}) \text{ if $l \geq i-1$}} \\ &= \cases{ e_l \quad \text{ if $l < i-1, l< j$}\\ e_{l+1} \ \text{ if $l < i-1, l \geq j$}\\ e_{l+1} \ \text{ if $l \geq i-1, l+1< j$}\\ e_{l+2} \ \text{ if $l \geq i-1, l+1 \geq j$}} \\ &= \cases{ e_l \quad \text{ if $l< j$}\\ e_{l+1} \ \text{ if $l < i-1, l \geq j$}\\ e_{l+2} \ \text{ if $l \geq i-1$}} \end{align} $$