theorem
  (S is discrete or S is op-discrete) implies A is monotone
proof
  assume
A1: S is discrete or S is op-discrete;
  let o1,o2;
  assume that
A2: o1 ~= o2 and
A3: (the_arity_of o1) <= (the_arity_of o2) & the_result_sort_of o1 <=
  the_result_sort_of o2;
  o1 = o2
  proof
    per cases by A1;
    suppose
      S is discrete;
      hence thesis by A2,A3,Th8;
    end;
    suppose
      S is op-discrete;
      hence thesis by A2,Th3;
    end;
  end;
  hence thesis;
end;
