
theorem Th48:
  for q be Element of REAL ex m be Element of INT st |.q - m .| <= 1/2
  proof
    let q be Element of REAL;
    per cases;
    suppose A1: |.q - [\q/] .| <= 1/2;
      reconsider  m= [\q/] as Element of INT by INT_1:def 2;
      take m;
      thus |.q - m .| <= 1/2 by A1;
    end;
    suppose A2: not |.q - [\q/] .| <= 1/2;
      0 <= q - [\q/] by INT_1:def 6,XREAL_1:48;
      then 1/2 < q - [\q/] by A2,ABSVALUE:def 1;
      then 1/2 -1 <= q - [\q/] -1  by XREAL_1:9;
      then A3: -1/2 <= q - ([\q/] + 1 );
      q - ([\q/] + 1) <= ([\q/] + 1) -([\q/] + 1) by INT_1:29,XREAL_1:9;
      then A4: q - ([\q/] + 1) <= 0;
      reconsider m = [\q/] + 1 as Element of INT by INT_1:def 2;
      take m;
      thus |. q - m  .| <= 1/2 by A3,A4,ABSVALUE:5;
    end;
  end;
