In this note we give an elementary proof of the strong normalization proper
ty of the J combinator by providing an explicit bound for the maximal lengt
h of the reduction paths of a term. This result shows clearly that in the t
heorem of Toyama, Klop and Barendregt on completeness of unions of left-lin
ear term rewriting systems, disjointness is essential. (C) 2001 Elsevier Sc
ience B.V. All rights reserved.