reserve x for Real;

theorem Th46:
  for z be Complex st Im z < 0 holds sin Arg z < 0
proof
  let z be Complex;
  Re z < 0 or Re z = 0 or Re z > 0;
  then
A1: Re z < 0 or Re z > 0 or z = (0+Im z*<i>) by COMPLEX1:13;
  assume Im z < 0;
  then
  Arg z in ].PI,3/2*PI.[ or Arg z in ].3/2*PI,2*PI.[ or Arg z = 3/2*PI by A1
,Th38,Th43,Th44;
  then
  PI < Arg z & Arg z < 3/2*PI or 3/2*PI < Arg z & Arg z < 2*PI or Arg z =
  3/2*PI by XXREAL_1:4;
  then PI < Arg z & Arg z < 2*PI by Lm5,Lm6,XXREAL_0:2;
  then Arg z in ].PI,2*PI.[ by XXREAL_1:4;
  then sin.Arg z < 0 by Th9;
  hence thesis by SIN_COS:def 17;
end;
