
theorem LM91:
  for d be Real holds |. sgn d .| <= 1
proof
  let d be Real;
  per cases;
  suppose d > 0; then
    sgn d = 1 by ABSVALUE:def 2;
    hence |. sgn d .| <= 1;
  end;
  suppose d < 0; then
    sgn d = -1 by ABSVALUE:def 2; then
    |. sgn d .| = -(-1) by ABSVALUE:def 1
               .= 1;
    hence |. sgn d .| <= 1;
  end;
  suppose d = 0; then
    sgn d = 0 by ABSVALUE:def 2;
    hence |. sgn d .| <= 1;
  end;
end;
