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 Th14:
  for Z being finite Tree st branchdeg (Root Z) = 2 holds succ (
  Root Z) = { <*0*>,<*1*> }
proof
  let Z be finite Tree;
  assume branchdeg (Root Z) = 2;
  then card succ (Root Z) = 2 by TREES_2:def 12;
  then consider x,y being object such that
A1: x <> y and
A2: succ (Root Z) = {x,y} by CARD_2:60;
A3: x in succ (Root Z) by A2,TARSKI:def 2;
  then reconsider x9 = x as Element of Z;
  x9 in { (Root Z)^<*n*> : (Root Z)^<*n*> in Z } by A3,TREES_2:def 5;
  then consider m such that
A4: x9 = (Root Z)^<*m*> and
  (Root Z)^<*m*> in Z;
A5: x9 = <*m*> by A4,FINSEQ_1:34;
A6: y in succ (Root Z) by A2,TARSKI:def 2;
  then reconsider y9 = y as Element of Z;
  y9 in { (Root Z)^<*n*> : (Root Z)^<*n*> in Z } by A6,TREES_2:def 5;
  then consider k such that
A7: y9 = (Root Z)^<*k*> and
  (Root Z)^<*k*> in Z;
A8: y9 = <*k*> by A7,FINSEQ_1:34;
  per cases;
  suppose
A9: m = 0;
    now
A10:  <*1*> = (Root Z)^<*1*> by FINSEQ_1:34;
      assume
A11:  k <> 1;
      then k <> 0 & ... & k <> 1 by A1,A5,A8,A9;
      then 1 < k by NAT_1:25;
      then 1+1 <= k by NAT_1:13;
      then <*1*> in Z by A8,Th3,XXREAL_0:2;
      then <*1*> in succ (Root Z) by A10,TREES_2:12;
      then <*1*> = <*0*> or <*1*> = <*k*> by A2,A5,A8,A9,TARSKI:def 2;
      hence contradiction by A11,TREES_1:3;
    end;
    hence thesis by A2,A4,A8,A9,FINSEQ_1:34;
  end;
  suppose
A12: m <> 0;
    <*0*> in Z & <*0*> = (Root Z)^<*0*> by A5,Th3,FINSEQ_1:34,NAT_1:2;
    then <*0*> in succ (Root Z) by TREES_2:12;
    then
A13: <*0*> = <*m*> or <*0*> = <*k*> by A2,A5,A8,TARSKI:def 2;
    now
A14:  <*1*> = (Root Z)^<*1*> by FINSEQ_1:34;
      assume
A15:  m <> 1;
      then 1 < m by A12,NAT_1:25;
      then 1+1 <= m by NAT_1:13;
      then <*1 *> in Z by A5,Th3,XXREAL_0:2;
      then <*1*> in succ (Root Z) by A14,TREES_2:12;
      then <*1*> = <*0*> or <*1*> = <*m*> by A2,A5,A8,A12,A13,TARSKI:def 2
,TREES_1:3;
      hence contradiction by A15,TREES_1:3;
    end;
    hence thesis by A2,A4,A8,A13,FINSEQ_1:34,TREES_1:3;
  end;
end;
