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 Th48:
  for T being finite-branching Tree st not T is finite ex C being
  Chain of T st not C is finite
proof
  let T be finite-branching Tree such that
A1: not T is finite;
A2: for n ex C being finite Chain of T st card C = n
  proof
    let n;
    T-level n <> {} by A1,Th47;
    then consider t being object such that
A3: t in T-level n by XBOOLE_0:def 1;
A4: ex w being Element of T st t = w & len w = n by A3;
    reconsider t as Element of T by A3;
    ProperPrefixes t is finite Chain of T by Th43;
    then consider C being finite Chain of T such that
A5: C = ProperPrefixes t;
    card C = n by A4,A5,TREES_1:35;
    hence thesis;
  end;
  for t being Element of T holds succ t is finite;
  hence thesis by A2,TREES_2:29;
end;
