I was trying to prove a binary operation is associative on a given number of the operand. I did it, but then I checked the textbook. I took 5 lines, but the book proved it in 6 lines using a precise step-by-step treatment. Here's the example: Let F denote the set of all real functions and define addition as the following \begin{equation} (f+g)(x)=f(x)+g(x) \quad \text { for each } x \in \mathbb{R} \end{equation} Now to prove it is associative: \begin{equation} \begin{aligned} ([f+(g+h)](x)) &=f(x)+(g+h)(x) & & \text { definition of }+\text { on } F \\ &=f(x)+[g(x)+h(x)] & & \text { definition of }+\text { on } F \\ &=[f(x)+g(x)]+h(x) & & \text { associativity of }+\text { on } \mathbb{R} \\ &=(f+g)(x)+h(x) & & \text { definition of }+\text { on } F \\ &=[(f+g)+h](x) & & \text { definition of }+\text { on } F \end{aligned} \end{equation}
After a quick search, I learned that the number of possible ways to insert parentheses is related to Catalan numbers, but this gives me the possible way to insert parenthesis, which has a different representation. But what I want is the number of steps I need to show that a binary operation is associative on a given number of operands. Any help?