reserve x for set,
  t,t1,t2 for DecoratedTree;

theorem
  t1 is finite & Subtrees t1 = Subtrees t2 implies t1 = t2
proof
  assume that
A1: t1 is finite and
A2: Subtrees t1 = Subtrees t2;
  reconsider t = t1 as finite DecoratedTree by A1;
  t1 in Subtrees t2 by A2,Th11;
  then consider n being Node of t2 such that
A3: t1 = t2|n;
  t2 in Subtrees t1 by A2,Th11;
  then consider m being Node of t1 such that
A4: t2 = t1|m;
  dom (t1|m) = (dom t1)|m by TREES_2:def 10;
  then reconsider p = m^n as Element of dom t by A4,TREES_1:def 6;
  t = t|p by A3,A4,Th3;
  then dom t = (dom t)|p by TREES_2:def 10;
  then n = {} by Th9;
  hence thesis by A3,Th1;
end;
