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

theorem Th8:
  for m,n,x,y being bag of I st n = (m-'x)+y holds m-'n divides x &
  n-'m divides y
  proof
    let m,n,x,y be bag of I;
    assume Z0: n = (m-'x)+y;
    thus m-'n divides x
    proof let a;
      n.a = (m-'x).a+y.a by Z0,PRE_POLY:def 5
      .= (m.a-'x.a)+y.a by PRE_POLY:def 6;
      then m.a-'n.a <= x.a by Th5;
      hence (m-'n).a <= x.a by PRE_POLY:def 6;
    end;
    let a;
    n.a = (m-'x).a+y.a by Z0,PRE_POLY:def 5
    .= (m.a-'x.a)+y.a by PRE_POLY:def 6;
    then n.a-'m.a <= y.a by Th5;
    hence thesis by PRE_POLY:def 6;
  end;
