Take a, b, c ∈ L. Obviously, because L is a linear ordered set, A, b and c must be comparable. Let a≤b≤c, then
a∩(b∩c)= a∪b = b，(a∪b)∩(a∪c)= b∪c = b .
b∩(a∩c)= b∪a = b，(b∪a)∩(b∪c)= b∪c = b .
c ∪( a∪b)= c∪a = c，(c∪a)∪( c∪b)= c∪c = c .
This proves the distribution from ∪ to ∩, and vice versa.