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"Don't panic!"

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I'm currently collating my own personal notes and would really appreciate some feedback on my description of the relativity of position and velocity in classical mechanics. Here is what I have written

"Position is clearly a relative quantity as two inertial frames [itex]S[/itex] and [itex]S'[/itex] displaced by a constant displacement vector [itex]\mathbb{r}_{0}[/itex] will measure the position of an object to be at [itex]\mathbb{r}[/itex] and [itex]\mathbb{r}'[/itex] respectively, the two positions related by [itex]\mathbb{r} = \mathbb{r}' + \mathbb{r}_{0}[/itex]. As these two frames are arbitrary and neither can be distinguished from the other as a preferred absolute rest frame (as a consequence of Galileo's principle of relativity), it must be that position is relative. This argument also holds if the two frames [itex]S[/itex] and [itex]S'[/itex] are in relative motion to one another, related by [itex]\mathbb{r} = \mathbb{r}'+\mathbb{v}t[/itex], where [itex]\mathbb{v}[/itex] is the relative velocity between the two frames. Clearly it follows from this (by differentiating with respect to time) that velocity is also relative."

"Position is clearly a relative quantity as two inertial frames [itex]S[/itex] and [itex]S'[/itex] displaced by a constant displacement vector [itex]\mathbb{r}_{0}[/itex] will measure the position of an object to be at [itex]\mathbb{r}[/itex] and [itex]\mathbb{r}'[/itex] respectively, the two positions related by [itex]\mathbb{r} = \mathbb{r}' + \mathbb{r}_{0}[/itex]. As these two frames are arbitrary and neither can be distinguished from the other as a preferred absolute rest frame (as a consequence of Galileo's principle of relativity), it must be that position is relative. This argument also holds if the two frames [itex]S[/itex] and [itex]S'[/itex] are in relative motion to one another, related by [itex]\mathbb{r} = \mathbb{r}'+\mathbb{v}t[/itex], where [itex]\mathbb{v}[/itex] is the relative velocity between the two frames. Clearly it follows from this (by differentiating with respect to time) that velocity is also relative."

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