
theorem Th8:
  for mlow,mhigh,f be Integer st mhigh < mlow + f & f > 0 holds ex
  s be Integer st -f < mlow - s * f & mhigh - s * f < f
proof
  let mlow,mhigh,f be Integer;
  assume that
A1: mhigh < mlow + f and
A2: f > 0;
  reconsider f as Element of NAT by A2,INT_1:3;
A3: mhigh + - ((mhigh div f) * f) < mlow + f + - ((mhigh div f) * f) by A1,
XREAL_1:6;
A4: mhigh mod f = mhigh - (mhigh div f) * f by A2,INT_1:def 10;
  then 0 <= mhigh - (mhigh div f) * f by A2,NAT_D:62;
  then 0 + -f < (mlow + - ((mhigh div f) * f)) + f + -f by A3,XREAL_1:6;
  then - f < mlow - (mhigh div f) * f;
  hence thesis by A2,A4,NAT_D:62;
end;
