
theorem Th26:
  for a being Complex, r being Real
   st r < 0 holds Arg(a*r) = Arg -a
proof
  let a be Complex, r be Real such that
A1: r < 0;
  per cases;
  suppose
    a = 0;
    hence thesis;
  end;
  suppose
A2: a <> 0;
    then
A3: cos Arg a = Re a/|.a.| by Th24;
A4: 0 <= Arg -a & Arg -a < 2*PI by COMPTRIG:34;
A5: sin Arg a = Im a/|.a.| by A2,Th24;
    set b = a*r;
A6: a in COMPLEX by XCMPLX_0:def 2;
A7: 0 <= Arg b & Arg b < 2*PI by COMPTRIG:34;
A8: |.b.| = |.a.|*|.r.| by COMPLEX1:65
      .= |.a.|*(-r) by A1,ABSVALUE:def 1;
    r =r+0*<i>;
    then
A9: Re r = r & Im r = 0 by COMPLEX1:12;
    then Im b = Re a * 0 + r * Im a by COMPLEX1:9
      .= r*Im a;
    then
A10: sin Arg b = r*Im a/(-|.a.|*r) by A1,A2,A8,Th24
      .= -r*Im a/( |.a.|*r) by XCMPLX_1:188
      .= -sin Arg a by A1,A5,XCMPLX_1:91
      .= sin Arg -a by A2,A6,Th23;
    Re b = Re a * r - 0*Im a by A9,COMPLEX1:9
      .= r*Re a;
    then cos Arg b = r*Re a/(-|.a.|*r) by A1,A2,A8,Th24
      .= -r*Re a/( |.a.|*r) by XCMPLX_1:188
      .= -cos Arg a by A1,A3,XCMPLX_1:91
      .= cos Arg -a by A2,A6,Th23;
    hence thesis by A10,A7,A4,Th11;
  end;
end;
