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;
reserve s9,w9,v9 for Element of NAT*;
reserve p,q for MP-variable;
reserve A,A1,B,B1,C,C1 for MP-wff;

theorem Th25:
  (elementary_tree 0) --> [3,n] is MP-wff
proof
  3 in NAT & n in NAT by ORDINAL1:def 12;
  then reconsider 3n = [3,n] as Element of [:NAT,NAT:] by ZFMISC_1:87;
  set x = (elementary_tree 0) --> 3n;
A1: dom x = { {} } by FUNCOP_1:13,TREES_1:29;
  reconsider x as finite DecoratedTree of [: NAT,NAT :];
A2: dom x = elementary_tree 0 by FUNCOP_1:13;
  for v being Element of dom x holds branchdeg v <= 2 & (branchdeg v = 0
implies x .v = [0,0] or ex k st x .v = [3,k]) & (branchdeg v = 1 implies x .v =
  [1,0] or x .v = [1,1]) & (branchdeg v = 2 implies x .v = [2,0])
  proof
    let v be Element of dom x;
A3: succ v = {}
    proof
      set y = the Element of succ v;
      assume not thesis;
      then y in succ v;
      then y in { v^<*m*> : v^<*m*> in dom x } by TREES_2:def 5;
      then ex m st y = v^<*m*> & v^<*m*> in dom x;
      hence contradiction by A1,TARSKI:def 1;
    end;
    hence branchdeg v <= 2 by CARD_1:27,TREES_2:def 12;
    thus branchdeg v = 0 implies x .v = [0,0] or ex m st x .v =[3,m] by A2,
FUNCOP_1:7;
    thus thesis by A3,CARD_1:27,TREES_2:def 12;
  end;
  hence thesis by Def5;
end;
