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 Th40:
  q is ordered & q = <*a*>^p implies a = b|{x:x is_maximal_in support b}
  proof
    assume Z0: q is ordered;
    assume Z1: q = <*a*>^p;
    let y; set J = {x:x is_maximal_in support b};
    per cases;
    suppose
A0:   y in J;
      then consider x such that
A1:   y = x & x is_maximal_in support b;
      a.y > 0 by Z0,Z1,A1,Th34;
      hence a.y = b.y by Z0,Z1,Th31 .= (b|J).y by A0,BAR;
    end;
    suppose
A2:   not y in J;
      then not y is_maximal_in support b;
      then a.y <= 0 by Z0,Z1,Th34;
      hence a.y = 0 .= (b|J).y by A2,BAR;
    end;
  end;
