
theorem
  for z being Complex st Arg z <> 0 holds Arg z < PI iff sin Arg z> 0
proof
  let z be Complex;
A1: Arg(z)>=0 by COMPTRIG:34;
  assume
A2: Arg(z)<>0;
  hereby
    assume Arg(z)<PI;
    then Arg(z) in ].0,PI.[ by A2,A1,XXREAL_1:4;
    then Im (z)>0 by Th16;
    hence sin (Arg(z))>0 by COMPTRIG:45;
  end;
A3: ].0,PI/2.[ c= ].0,PI.[ by COMPTRIG:5,XXREAL_1:46;
  thus sin (Arg(z))>0 implies Arg(z)<PI
  proof
    assume
A4: sin (Arg(z))>0;
    then
A5: Im (z)>0 by COMPTRIG:48;
    now
      per cases;
      suppose
        Re z>0;
        then Arg(z) in ].0,PI/2.[ by A5,COMPTRIG:41;
        hence thesis by A3,XXREAL_1:4;
      end;
      suppose
        Re z=0;
        then z= 0 + (Im z)*<i> by COMPLEX1:13;
        hence thesis by A4,COMPTRIG:5,37,48;
      end;
      suppose
        Re z<0;
        then Arg(z) in ].PI/2,PI.[ by A5,COMPTRIG:42;
        hence thesis by XXREAL_1:4;
      end;
    end;
    hence thesis;
  end;
end;
