theorem Th20:
  the_arity_of o = {} implies height (o-term p) = 0
  proof
    assume the_arity_of o = {};
    then dom p = dom {} by MSUALG_3:6;
    then p = {};
    then o-term p = root-tree [o,the carrier of S] by TREES_4:20;
    hence thesis by TREES_1:42;
  end;
