# Distributive lattices and Birkhoff theorem

I am trying to prove the teorem

(Birkhoff) $L$ is a nondistributive lattice iff $M_5$ or $N_5$ can be embedded into $L$

The only part of the proof which I can't understand is this (I am copying from my book): suppose $L$ nondistributive. Thus there must be elements $a,b,c$ in $L$ such that $(a\wedge b)\vee (a\wedge c) < a \wedge (b\vee c)$. Let us define

$d:=(a\wedge b)\vee (a\wedge c) \vee (b\wedge c)$

$e:=(a\vee b)\wedge (a\vee c)\wedge (b\vee c)$

$a_1:=(a\wedge e)\vee d$

$b_1:=(b\wedge e)\vee d$

$c_1=(c\wedge e)\vee d$

Then it is easily seen that $d\leq a_1,b_1,c_1\leq e$ ......et cetera et cetera...

What I can't get is the last claim: why $a_1$ (and similarly $b_1$ and $c_1$) are $\leq$ than $e$?

Hint: you know that $d \leqslant e$. What can you say about $a\wedge e$?
• that $a\wedge d\leq a\wedge e$ – Danae Kissinger Jul 17 '14 at 18:45
• $d\leq e$ and $a\wedge e\leq e$, thus $(a\wedge e) \vee d\leq e$, ok i got it. Thanks. – Danae Kissinger Jul 17 '14 at 19:07