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;

theorem
  r - 1 < [/ r \] & r < [/ r \] + 1
proof
  r <= [/ r \] by Def7;
  then
A1: r + 0 < [/ r \] + 1 by XREAL_1:8;
  then r + (- 1) < [/ r \] + 1 + (- 1) by XREAL_1:6;
  hence thesis by A1;
end;
