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