theorem
  x in I & (for y st y in I & y <> x holds x ## y) implies x is_maximal_in I
  proof
    assume Z0: x in I;
    assume Z1: for y st y in I & y <> x holds x ## y;
    not ex y st y in I & x < y
    proof let y;
      assume y in I & x <= y & x <> y;
      then x ## y & x <= y by Z1;
      hence contradiction;
    end;
    hence x is_maximal_in I by Z0,WAYBEL_4:55;
  end;
