theorem
  n > 0 & k mod n <> 0 implies - (k div n) = (-k) div n + 1
proof
  assume
A1: n > 0;
  assume k mod n <> 0;
  then not n qua Integer divides k by A1,INT_1:62;
  then
A2: k/n is not Integer by A1,Th17;
  thus - (k div n) = - [\ k / n /] by INT_1:def 9
    .= [\ (-k) / n /] + 1 by A2,INT_1:63
    .= (-k) div n + 1 by INT_1:def 9;
end;
