Mapping/Composition/Associativity/Fact/Proof/Exercise/Solution

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

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

To the exercise