reserve x for Real;

theorem Th49:
  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 ].0,PI/2.[ or Arg z in ].3/2*PI,2*PI.[ or Arg z = 0 by A1,Th35
,Th41,Th44;
  then cos.Arg z > 0 by Th15,SIN_COS:30,80;
  hence thesis by SIN_COS:def 19;
end;
