Is the Reversion map in Geometric Algebra well-defined?

Question

I am studying from the book "Geometric Algebra for Physicists: by Chris Doran and Anthony Lasenby." In the book they define a map $\dagger : \mathcal G \to \mathcal G$, where $\mathcal G$ is a geometric algebra, by $(a_1\ldots a_r)^\dagger=a_r\ldots a_1$ where $a_1,\ldots,a_r$ are vectors. My question is if $$a_1\ldots a_r=b_1 \ldots b_s$$ where $r$ is not necessarily equal to $s$ how does one know that $(a_1\ldots a_r)^\dagger=(b_1\ldots b_s)^\dagger$?