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The proof is by contradiction. So assume that \(K_5\) is planar. Then the graph must satisfy Euler's formula for planar graphs. \(K_5\) has 5 vertices and 10 edges, so we get

\begin{equation*} 5 - 10 + f = 2\text{,} \end{equation*}

which says that if the graph is drawn without any edges crossing, there would be \(f = 7\) faces.

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