
theorem Th23:
  for z being Element of COMPLEX st z <> 0 holds cos Arg -z = -cos
  Arg z & sin Arg -z = - sin Arg z
proof
  let a be Element of COMPLEX;
A1: |.-a.| = |.a.| by COMPLEX1:52;
  assume a <> 0;
  then
A2: |.a.|<>0 by COMPLEX1:45;
  a = |.a.|*cos Arg a+(|.a.|*sin Arg a)*<i> & -a = |.-a.|*cos Arg -a+(|.-a
  .|* sin Arg -a)*<i> by COMPTRIG:62;
  then
A3: 0+0*<i> = |.a.|*cos Arg a + (|.a.|*cos Arg -a)+ (|.a.|*sin Arg a + (|.a
  .|*sin Arg -a))*<i> by A1;
  then |.a.|*(sin Arg a+sin Arg -a)= 0 by COMPLEX1:4,12;
  then
A4: sin Arg a +--sin Arg -a = 0 by A2;
  |.a.|*(cos Arg a + cos Arg -a) = 0 by A3,COMPLEX1:4,12;
  then cos Arg a +--cos Arg -a = 0 by A2;
  hence thesis by A4;
end;
