Why in distributive lattices, if an element has complement, it is unique?

Question

I know in some lattice like Diamond, a given element can have more than one complement, but when we have a distributive lattice, an element has at most one complement. I’m looking for prove of this Theorem, please help me.

Logically in Lattice L (Inf is Infimum and Sup is Supremum):

∃x,y: (Inf(a,x)=0 ^ sup(a,x)=1) ^ (inf(a,y)=0 ^ Sup(a,y)=1)

If L is distributive we can prove x=y.

(Am i right?)

P.S. It’s very helpful if you describe Line Of Reasoning of The prove too.

Thanks, Omid Yaghoubi

Answer 1

Suppose that $x$ and $y$ are two complements of $a$. We have:

$x = x \vee 0 = x \vee (a\wedge y) = (x\vee a)\wedge (x \vee y) = 1\wedge (x \vee y) = x\vee y$.

Similarly, $y = x\vee y$, so $x=y$.