# Proof about number of vertices ($\mathbb{R}^3$ space)…

I do not fully understand the proof in Wikipedia, the first paragraph of this.

$$\begin{cases} v − e + f = 2 \\ 2e \ge 3f \end{cases}$$

Firstly $2e \ge 3f$ means that a facet has at least 3 edges, at the same time one edge exists besides exactly 2 facets - I guess that's how it is, right?

The conclusion $v \le 2f − 4$ however is still unclear. They simply took a legit $f \le 2v − 4$ and exchanged the variables. How is that allowed?

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–  user7530 Feb 19 '13 at 17:35

I'd love to come up with the same property for $\mathbb{R}^4$. Do you think that's possible? Given that the above applies only to vertices-facets... –  Ranas Feb 19 '13 at 17:58