reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;
reserve C for Chain of W,
  B for Branch of W;
reserve T,T1,T2 for DecoratedTree;
reserve x1,x2 for set,
  w for FinSequence of NAT;

theorem
  for T1, T2 being Tree st
     for n being Nat holds T1-level n = T2-level n
   holds T1 = T2
proof
  let T1, T2 be Tree such that
A1: for n being Nat holds T1-level n = T2-level n;
  for p being FinSequence of NAT holds p in T1 iff p in T2
  proof
    let p be FinSequence of NAT;
A2: T1 = union the set of all  T1-level n where n is Nat
    by Th14;
    hereby
      assume p in T1;
      then consider Y being set such that
A3:   p in Y and
A4:   Y in the set of all  T1-level n where n is Nat
      by A2,TARSKI:def 4;
      consider n being Nat such that
A5:   Y = T1-level n by A4;
      Y = T2-level n by A1,A5;
      hence p in T2 by A3;
    end;
    assume
A6: p in T2;
    T2 = union the set of all  T2-level n where n is Nat
    by Th14;
    then consider Y being set such that
A7: p in Y and
A8: Y in the set of all  T2-level n where n is Nat
    by A6,TARSKI:def 4;
    consider n being Nat such that
A9: Y = T2-level n by A8;
    Y = T1-level n by A1,A9;
    hence thesis by A7;
  end;
  hence thesis;
end;
