reserve T, T1 for Tree,
  P for AntiChain_of_Prefixes of T,
  p1 for FinSequence,
  p, q, r, s, p9 for FinSequence of NAT,
  x, Z for set,
  t for Element of T,
  k, n for Nat;

theorem Th1:
  for p,q,r,s being FinSequence st p^q = s^r holds p,s are_c=-comparable
proof
  let p,q,r,s be FinSequence;
  assume
A1: p^q = s^r;
  then p is_a_prefix_of s^r & s is_a_prefix_of p^q by TREES_1:1;
  hence thesis by A1,TREES_2:1;
end;
