theorem Th22:
  Z c= X & Z<>{} & X common_on_dom H implies Z common_on_dom H
  proof
    assume that
    A1: Z c= X and
    A2: Z<>{} and
    A3: X common_on_dom H;
    now
      let n;
      X c= dom (H.n) by A3;
      hence Z c= dom (H.n) by A1;
    end;
    hence thesis by A2;
  end;
