Homotopy between composites associated in different ways

Existential version
Suppose $$f_1,f_2,f_3$$ are loops based at a point $$x_0$$ in a topological space $$X$$. We can consider two differently associated products of these three loops:

$$\! a = (f_1 * f_2) * f_3, b = f_1 * (f_2 * f_3)$$

$$a$$ and $$b$$ are homotopic loops, i.e., they are in the same homotopy class of loops based at $$x_0$$.

This version is essentially the associativity part of showing that the fact about::fundamental group of a based topological space is indeed a group.

Constructive/explicit version
We first note the explicit piecewise definitions of $$a$$ and $$b$$:

$$a = \lbrace\begin{array}{rl} f_1(4t), & 0 \le t \le 1/4 \\ f_2(4t - 1), & 1/4 < t \le 1/2 \\ f_3(2t - 1), & 1/2 < t \le 1 \\\end{array}$$

and:

$$b = \lbrace\begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(4t - 2), & 1/2 < t \le 3/4 \\ f_3(4t - 3), & 3/4 < t \le 1 \\\end{array}$$

If we denote the homotopy by $$H$$, we want $$H(t,0) = a(t), H(t,1) = b(t)$$ and $$H(0,s) = H(1,s) = x_0$$. This homotopy is explicitly given by:

$$H(t,s) = \lbrace\begin{array}{rl} f_1\left(\frac{4t}{1 + s}\right), & 0 \le t \le (1 + s)/4 \\ f_2(4t - 1 - s), & (1 + s)/4 < t \le (2 + s)/4 \\ f_3\left(\frac{4t - 2 - s}{2 - s}\right), & (2 + s)/4 < t \le 1 \\\end{array}$$

Uniform version
This version is a little stronger than the other versions. Let $$L = \Omega(X,x_0)$$ be the loop space of $$(X,x_0)$$, i.e., the space of all loops in $$X$$ based at $$x_0$$ under the compact-open topology. Then, consider the following two maps:

$$\! A:L \times L \times L \to L, \qquad A(f_1,f_2,f_3) = (f_1 * f_2) * f_3$$

and:

$$\! B:L \times L \times L \to L, \qquad B(f_1,f_2,f_3) = f_1 * (f_2 * f_3)$$

Then, the maps $$A$$ and $$B$$ are homotopic maps. This is part of the proof that $$\Omega(X,x_0)$$ is a H-space, which is a homotopy variant of topological monoid.