Mapping/Composition/Associativity/Fact/Proof

Two mappings ${\displaystyle {}\alpha ,\beta \colon L\rightarrow P}$ are the same if and only if the equality ${\displaystyle {}\alpha (x)=\beta (x)}$ holds for every ${\displaystyle {}x\in L}$. So let ${\displaystyle {}x\in L}$. Then

${\displaystyle {}{\begin{aligned}(H\circ (G\circ F))(x)&=H((G\circ F)(x))\\&=H(G(F(x)))\\&=(H\circ G)(F(x))\\&=((H\circ G)\circ F)(x).\end{aligned}}}$