reserve x for Real;

theorem Th50:
  for z be Complex st Re z < 0 holds cos Arg z < 0
proof
  let z be Complex;
  Im z < 0 or Im z = 0 or Im z > 0;
  then
A1: Im z < 0 or Im z > 0 or z = (Re z+0*<i>) by COMPLEX1:13;
  assume Re z < 0;
  then Arg z in ].PI/2,PI.[ or Arg z in ].PI,3/2*PI.[ or Arg z = PI by A1,Th36
,Th42,Th43;
  then PI/2 < Arg z & Arg z < PI or PI < Arg z & Arg z < 3/2*PI or Arg z = PI
  by XXREAL_1:4;
  then PI/2 < Arg z & Arg z < 3/2*PI by Lm2,Lm5,XXREAL_0:2;
  then Arg z in ].PI/2,3/2*PI.[ by XXREAL_1:4;
  then cos.Arg z < 0 by Th13;
  hence thesis by SIN_COS:def 19;
end;
