reserve x,y,x1,x2,z for set,
  n,m,k for Nat,
  t1 for (DecoratedTree of [: NAT,NAT :]),
  w,s,t,u for FinSequence of NAT,
  D for non empty set;

theorem Th12:
  for Z being finite Tree st branchdeg (Root Z) = 0 holds card Z = 1 & Z = {{}}
proof
  let Z be finite Tree;
  assume branchdeg (Root Z) = 0;
  then 0 = card succ (Root Z) by TREES_2:def 12;
  then
A1: succ (Root Z) = {};
  now
    let x be object;
    thus x in Z implies x in { Root Z }
    proof
      assume x in Z;
      then reconsider z = x as Element of Z;
      assume not thesis;
      then z <> Root Z by TARSKI:def 1;
      then consider w being FinSequence of NAT,
      n being Element of NAT such that
A2:   z = <*n*>^w by FINSEQ_2:130;
      <*n*> is_a_prefix_of z by A2,TREES_1:1;
      then <*n*> in Z by TREES_1:20;
      then {}^<*n*> in Z by FINSEQ_1:34;
      hence contradiction by A1,TREES_2:12;
    end;
    assume x in { Root Z };
    then reconsider x9= x as Element of Z;
    x9 in Z;
    hence x in Z;
  end;
  then Z = { Root Z } by TARSKI:2;
  hence thesis by CARD_2:42;
end;
