reserve x, y for set;

theorem Th4:
  for a being Integer for b being Real st |. a - b .| < 1/2 holds
  a = round b
  proof
    let a be Integer;
    let b be Real;
    assume
A1: |. a - b .| < 1/2;
    then a - b < 1/2 by SEQ_2:1; then
A2: a - b + b < 1/2 + b by XREAL_1:8;
    -1/2 < a - b by A1,SEQ_2:1;
    then -(a - b) < -(-1/2) by XREAL_1:24;
    then b - a + a < 1/2 + a by XREAL_1:8;
    then b - 1/2 < a + 1/2 - 1/2 by XREAL_1:14;
    then b + 1/2 - 1 < a;
    hence thesis by A2,INT_1:def 6;
  end;
