reserve x, r for Real;
reserve A for symmetrical Subset of COMPLEX;
reserve F,G for PartFunc of REAL, REAL;

theorem Th59:
  for x being Real holds signum.(-x) = -signum.x
proof
  let x be Real;
  per cases;
  suppose
A1: x < 0;
    then signum.x = -1 by Th57;
    hence thesis by A1,Th56;
  end;
  suppose
A2: 0 < x;
    then signum.x = 1 by Th56;
    hence thesis by A2,Th57;
  end;
  suppose x = 0;
    hence thesis by Th58;
  end;
end;
