
theorem DIV:
  for a be Integer, b be non zero Integer st
    not b divides a holds (a div b) + ((-a) div b) = -1
  proof
    let a be Integer, b be non zero Integer such that
    A1: not b divides a;
    reconsider a as non zero Integer by A1,INT_2:12;
    A2: a div b = [\a/b/] & (-a) div b = [\(-a)/b/] by INT_1:def 9;
    (a/b)*b = a by XCMPLX_1:87; then
    not a/b is Integer by A1; then
    -[\a/b/] = [\(-a)/b/] + 1 by INT_1:63; then
    [\a/b/] + [\(-a)/b/] = -1;
    hence thesis by A2;
  end;
