
theorem Th20:
  for z being Complex holds Arg z = PI iff Re z < 0 & Im z= 0
proof
  let z be Complex;
  hereby
    assume
A1: Arg z = PI;
    per cases;
    suppose
A2:    z<>0;
     reconsider zz=|.z.| as Element of REAL by XREAL_0:def 1;
A3:  z=zz*cos PI+|.z.|*sin PI *<i> & --|.z.| > 0
      by A1,COMPLEX1:47,COMPTRIG:def 1,A2;
   cos PI = -1 & sin PI = 0 by SIN_COS:77;
      then
A5:    z =zz*(-1)+zz*0*<i> by A3;
     hence Re z < 0 by COMPLEX1:def 1,A3;
     thus Im z = 0 by COMPLEX1:def 2,A5;
    end;
    suppose
      z=0;
      hence Re z < 0 & Im z = 0 by A1,COMPTRIG:5,35;
    end;
  end;
  assume that
A6: Re z < 0 and
A7: Im z=0;
  z = Re z +0*<i> by A7,COMPLEX1:13;
  hence thesis by A6,COMPTRIG:36;
end;
