nonmodular sublattice

Any nonmodular lattice $L$ contains the lattice $N_5$ (shown below) as a sublattice.

Proof.

Since $L$ is not modular, by definition it contains elements $x$, $y$ and $z$ such that $x\le z$ and . Then the sublattice formed by $y$, $x\vee y$, $y\wedge z$, $(x\vee y)\wedge z$ and $x\vee (y\wedge z)$ is isomorphic to $N_5$. This is because while $[x\vee (y\wedge z)]\vee y=x\vee y$ by absorption and similarly $y\wedge [(x\vee y)\wedge z]=y\wedge z$. Moreover, $x\vee (y\wedge z)$ covers $y\wedge z$ since $y\wedge z=x\vee (y\wedge z)$ would imply $x\le y\wedge z\le y$, whence $x\vee y=y$ and $(x\vee y)\wedge z=y\wedge z=x\vee (y\wedge z)$ contrary to our hypothesis. By the same method, $(x\vee y)\wedge z=x\vee y$ leads to a contradiction of nonmodularity, so $x\vee y$ covers $(x\vee y)\wedge z$.∎