reserve a,b for object, I,J for set;

theorem Th9:
  for m,x,y being bag of I st x divides m & x <> y holds m <> (m-'x)+y
  proof
    let m,x,y be bag of I;
    assume Z0: for a holds x.a <= m.a;
    given a such that
Z1: a in I & x.a <> y.a;
    take a; thus a in I by Z1;
A1: ((m-'x)+y).a = (m-'x).a+y.a = (m.a-'x.a)+y.a by PRE_POLY:def 5,def 6;
    m.a-'x.a = m.a-x.a by Z0,XREAL_1:233;
    hence thesis by A1,Z1;
  end;
