reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;

theorem
  Subtrees {t} = Subtrees t
proof
  hereby
    let x be object;
    assume x in Subtrees {t};
    then consider u being Element of {t}, p being Node of u such that
A1: x = u|p;
    u = t by TARSKI:def 1;
    hence x in Subtrees t by A1;
  end;
  let x be object;
  assume x in Subtrees t;
  then t in {t} & ex p being Node of t st x = t|p by TARSKI:def 1;
  hence thesis;
end;
