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

theorem
  X c= Y implies Subtrees X c= Subtrees Y
proof
  assume
A1: for x being object holds x in X implies x in Y;
  let x be object;
  assume x in Subtrees X;
  then consider t being Element of X, p being Node of t such that
A2: x = t|p;
  reconsider t as Element of Y by A1;
  reconsider p as Node of t;
  x = t|p by A2;
  hence thesis;
end;
