reserve x for Real;

theorem
  for z be Complex st Re z >= 0 holds cos Arg z >= 0
proof
  let z be Complex;
  assume Re z >= 0;
  then Re z > 0 or Re z = 0;
  then
  cos Arg z > 0 or z = (0+Im z*<i>) & (Im z > 0 or Im z < 0 or Im z = 0 )
  by Th49,COMPLEX1:13;
  hence thesis by Def1,Th37,Th38,SIN_COS:31,77;
end;
