reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;
reserve T for DecoratedTree,
  p for FinSequence of NAT;
reserve T for finite-branching DecoratedTree,
  t for Element of dom T,
  x for FinSequence,
  n, m for Nat;
reserve x, x9 for Element of dom T,
  y9 for set;
reserve n,k1,k2,l,k,m for Nat,
  x,y for set;

theorem Th43:
  for T being Tree, t being Element of T holds ProperPrefixes t is
  finite Chain of T
proof
  let T be Tree, t be Element of T;
  ProperPrefixes t c= T & for p,q being FinSequence of NAT st p in
ProperPrefixes t & q in ProperPrefixes t holds p,q are_c=-comparable by
TREES_1:18,def 3;
  hence thesis by TREES_2:def 3;
end;
