reserve a,b for object, I,J for set;
reserve b for bag of I;
reserve R for asymmetric transitive non empty RelStr,
  a,b,c for bag of the carrier of R,
  x,y,z for Element of R;
reserve p for partition of b-'a, q for partition of b;
reserve J for set, m for bag of I;

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;
