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 $a\wedge e$?
