
theorem TA1:
  for a be Real holds
  (a is heavy negative iff a < -1) &
  (a is light negative iff -1 < a < 0) &
  (a is light positive iff 0 < a < 1) &
  (a is heavy positive iff a > 1) &
  (a is weightless positive iff a = 1) &
  (a is weightless negative iff a = -1)
  proof
    let a be Real;
    A1: a is heavy negative implies a < -1
    proof
      assume a is heavy negative; then
      -a > 1 by ABSVALUE:def 1; then
      -(-a) < -1 by XREAL_1:24;
      hence thesis;
    end;
    A1a: a < -1 implies a is heavy negative
    proof
      assume
      B1: a < -1; then
      -a > -(-1) by XREAL_1:24;
      hence thesis by B1,ABSVALUE:def 1;
    end;
    A3: a is light negative implies -1 < a < 0
    proof
      assume
      B1: a is light negative; then
      -a < 1 by ABSVALUE:def 1; then
      -(-a) > -1 by XREAL_1:24;
      hence thesis by B1;
    end;
    A3a: -1 < a < 0 implies a is light negative
    proof
      assume
      B1: -1 < a < 0; then
      -a < -(-1) by XREAL_1:24;
      hence thesis by B1,ABSVALUE:def 1;
    end;
    a is weightless implies a = 0 or 1 = a or 1 = -a by ABSVALUE:1;
    hence thesis by A1,A1a,ABSVALUE:def 1,A3,A3a;
  end;
