reserve x for Real;

theorem
  for z be Complex st Re z <= 0 & z <> 0 holds cos Arg z <= 0
proof
  let z be Complex;
  assume that
A1: Re z <= 0 and
A2: z <> 0;
A3: 0 = 0+0*<i>;
  Re z < 0 or Re z = 0 by A1;
  then cos Arg z < 0 or z = 0+Im z*<i> & (Im z >= 0 or Im z < 0) by Th50,
COMPLEX1:13;
  then cos Arg z < 0 or z = 0+Im z*<i> & (Im z > 0 or Im z < 0) by A2,A3;
  hence thesis by Th37,Th38,SIN_COS:77;
end;
