reserve a, b, k, n, m for Nat,
  i for Integer,
  r for Real,
  p for Rational,
  c for Complex,
  x for object,
  f for Function;

theorem Th5:
  m / n = m div n implies m mod n = 0
proof
  assume
A1: m / n = m div n;
  per cases;
  suppose
    n = 0;
    hence thesis;
  end;
  suppose
A2: n > 0;
    then m + 0 = n * (m/n) + (m mod n) by A1,NAT_D:2;
    then m mod n - 0 = m - n * (m/n);
    hence m mod n = m - m by A2,XCMPLX_1:87
      .= 0;
  end;
end;
