Left division in the free left distributive algebra on one generator

Let A be the free algebra on one generator satisfying the left distributive law a(bc)=(ab)(ac). Using a division algorithm for elements of an extension P of A, we prove some facts about left division in A, one consequence of which is a conjecture of J. Moody: If a,b,c,d∈A,ab=cd,a and b have no common left divisors, and c and d have no common left divisors, then a=c and b=d.