reserve x for Real;

theorem
  for z be Complex st Im z <= 0 holds sin Arg z <= 0
proof
  let z be Complex;
  assume Im z <= 0;
  then Im z < 0 or Im z = 0;
  then sin Arg z < 0 or z = (Re z+0*<i>) & (Re z >= 0 or Re z < 0) by Th46,
COMPLEX1:13;
  hence thesis by Th35,Th36,SIN_COS:31,77;
end;
