reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;
reserve C for Chain of W,
  B for Branch of W;

theorem Th27:
  p in C & q in C & len p <= len q implies p is_a_prefix_of q
proof
  assume p in C & q in C & len p <= len q & not p is_a_prefix_of q;
  then  q
 in ProperPrefixes p & not q is_a_proper_prefix_of p by Th21,TREES_1:6;
  hence contradiction by TREES_1:12;
end;
