reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;
reserve r1,p,p1,g,g1,g2 for Real,
  Y for Subset of REAL;
reserve r, s for Real;

theorem
  r/s is not Integer implies - [\ r/s /] = [\ (-r) / s /] + 1
proof
  r/s - 1 < [\ r/s /] by Def6;
  then - (r/s - 1) > - [\ r/s /] by XREAL_1:24;
  then -r/s + 1 > - [\ r/s /];
  then - [\ r/s /] <= (-r) / s + 1 by XCMPLX_1:187;
  then
A1: - [\ r/s /] - 1 <= (-r) / s + 1 - 1 by XREAL_1:9;
  assume r/s is not Integer;
  then [\ r/s /] < r/s by Th26;
  then -r/s < - [\ r/s /] by XREAL_1:24;
  then -r/s - 1 < - [\ r/s /] - 1 by XREAL_1:9;
  then (-r)/s - 1 < - [\ r/s /] - 1 by XCMPLX_1:187;
  then - [\ r/s /] - 1 + 1 = [\ (-r) / s /] + 1 by A1,Def6;
  hence thesis;
end;
