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;

theorem Th1:
  for v1,v2,v st v1 is_a_prefix_of v & v2 is_a_prefix_of v
  holds v1,v2 are_c=-comparable
proof
  let p,q,r be FinSequence;
  assume p is_a_prefix_of r;
then A1: ex p9 being FinSequence st r = p^p9 by TREES_1:1;
  assume q is_a_prefix_of r;
then A2: ex q9 being FinSequence st r = q^q9 by TREES_1:1;
 len p <= len q or len q <= len p;
then  (ex t being FinSequence st p^t = q) or
  ex t being FinSequence st q^t = p by A1,A2,FINSEQ_1:47;
  hence p is_a_prefix_of q or q is_a_prefix_of p by TREES_1:1;
end;
