theorem
  [\ r /] - 1 < r & [\ r /] < r + 1
proof
  [\ r /] <= r by Def6;
  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;
