reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th56:
  Rotate(-a,r) = -Rotate(a,r)
proof
  per cases;
  suppose
A1: a <> 0c;
A2: cos (r+Arg-a) = -cos(r+Arg a) & sin (r+Arg-a) = -sin(r+Arg a)
    proof
      per cases;
      suppose
        Arg a < PI;
        then Arg -a = PI+Arg a by A1,Th12;
        then r+Arg-a = PI+(r+Arg a);
        hence thesis by SIN_COS:79;
      end;
      suppose
        Arg a >= PI;
        then Arg -a = (Arg a)-PI by A1,Th12;
        then r+Arg-a = (Arg a)+r-PI;
        hence thesis by Th5;
      end;
    end;
    |.a.| = |.-a.| by COMPLEX1:52;
    hence thesis by A2;
  end;
  suppose
A3: a = 0c;
    hence Rotate(-a,r) = -0 by Th50
      .= -Rotate(a,r) by A3,Th50;
  end;
end;
