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 Th49:
  for T being finite-branching Tree st not T is finite ex B being
  Branch of T st not B is finite
proof
  let T be finite-branching Tree;
  assume not T is finite;
  then consider C being Chain of T such that
A1: not C is finite by Th48;
  consider B being Branch of T such that
A2: C c= B by TREES_2:28;
  not B is finite by A1,A2;
  hence thesis;
end;
