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 Th42:
  a <> EmptyBag the carrier of R implies {x:x is_maximal_in support a} <> {}
  proof set I = the carrier of R;
    given x such that
A1: a.x <> (EmptyBag I).x;
    (EmptyBag I).x = 0 by FUNCOP_1:7;
    then x in support a by A1,PRE_POLY:def 7;
    then consider x such that
A2: x is_maximal_in support a by Th12;
    x in {y:y is_maximal_in support a} by A2;
    hence thesis;
  end;
