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Proof

(related to Proposition: Composition of Functions is Associative)

• Let $f:A\mapsto B,$ $g:B\mapsto C,$ and $h:C\mapsto D$ be functions.
• The lemma defining compositions shows that both compositions $(h\circ g)\circ f$ and $h\circ (g\circ f)$ are functions.
• Moreover, these functions are identical, since for every $a\in A$ we have $$(h\circ g)\circ f(a)=h(g\circ f(a))=h(g(f(a)))=h\circ g(f(a))=h\circ(g\circ f)(a).$$
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References

Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
2. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
3. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994