
theorem Th25:
  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: sin Arg a = Im a/|.a.| by Th24;
    set b = a*r;
A4: |.b.| = |.a.|*|.r.| by COMPLEX1:65
      .= |.a.|*r by A1,ABSVALUE:def 1;
A5: cos Arg a = Re a/|.a.| by A2,Th24;
A6: 0 <= Arg a & Arg a < 2*PI by COMPTRIG:34;
    r=r+0*<i>;
    then
A7: Re r = r & Im r = 0 by COMPLEX1:12;
    then
A8: Im b = Re a * 0 + r * Im a by COMPLEX1:9
      .= r*Im a;
A9: sin Arg b = Im b /|.b.| by A1,A2,Th24
      .=sin Arg a by A1,A8,A4,A3,XCMPLX_1:91;
A10: 0 <= Arg b & Arg b < 2*PI by COMPTRIG:34;
A11: Re b = Re a * r - 0*Im a by A7,COMPLEX1:9
      .= r*Re a;
    cos Arg b = Re b/|.b.|by A1,A2,Th24
      .=cos Arg a by A1,A11,A4,A5,XCMPLX_1:91;
    hence thesis by A9,A10,A6,Th11;
  end;
end;
