
theorem
  for a be Real holds (a is non light negative iff a <= -1) &
  (a is non heavy negative iff -1 <= a < 0) &
  (a is non heavy positive  iff 0 < a <= 1) &
  (a is non light positive iff 1 <= a)
  proof
    let a be Real;
    A1: a is non light negative implies a <= -1
    proof
      assume a is non light negative; then
      -a >= 1 by ABSVALUE:def 1; then
      -(-a) <= -1 by XREAL_1:24;
      hence thesis;
    end;
    A1a: a <= -1 implies a is non light negative
    proof
      assume
      B1: a <= -1; then
      -a >= -(-1) by XREAL_1:24;
      hence thesis by B1,ABSVALUE:def 1;
    end;
    A3: a is non heavy negative implies -1 <= a < 0
    proof
      assume
      B1: a is non heavy negative; then
      -a <= 1 by ABSVALUE:def 1; then
      -(-a) >= -1 by XREAL_1:24;
      hence thesis by B1;
    end;
    -1 <= a < 0 implies a is non heavy negative
    proof
      assume
      B1: -1 <= a < 0; then
      -a <= -(-1) by XREAL_1:24;
      hence thesis by B1,ABSVALUE:def 1;
    end;
    hence thesis by A1,A1a,A3;
  end;
