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 Th13:
  for Z being finite Tree st branchdeg (Root Z) = 1 holds succ (
  Root Z) = { <*0*> }
proof
  let Z be finite Tree;
  assume branchdeg (Root Z) = 1;
  then card succ (Root Z) = 1 by TREES_2:def 12;
  then consider x being object such that
A1: succ (Root Z) = {x} by CARD_2:42;
A2: x in succ (Root Z) by A1,TARSKI:def 1;
  then reconsider x9 = x as Element of Z;
  x9 in { (Root Z)^<*n*> : (Root Z)^<*n*> in Z } by A2,TREES_2:def 5;
  then consider m such that
A3: x9 = (Root Z)^<*m*> and
A4: (Root Z)^<*m*> in Z;
A5: x9 = <*m*> by A3,FINSEQ_1:34;
  now
A6: <*0*> = (Root Z)^<*0*> by FINSEQ_1:34;
    <*m*> in Z by A4,FINSEQ_1:34;
    then <*0*> in Z by Th3,NAT_1:2;
    then (Root Z)^<*0*> in succ (Root Z) by A6,TREES_2:12;
    then
A7: <*0*> = x by A1,A6,TARSKI:def 1;
    assume m <> 0;
    hence contradiction by A5,A7,TREES_1:3;
  end;
  hence thesis by A1,A3,FINSEQ_1:34;
end;
