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;
