Homotopy between composites associated in different ways

Statement

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 $x_{0}$.

Constructive/explicit version

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