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

Two mappings $\alpha ,\beta :L\to P$ 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

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

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