reserve x for Real;

theorem
  for z be Complex holds Im z >= -|.z.|
proof
  let z be Complex;
  0 <= (Re z)^2 by XREAL_1:63;
  then
A1: (Im z)^2+0 <= (Re z)^2 + (Im z)^2 by XREAL_1:7;
  0 <= (Im z)^2 by XREAL_1:63;
  then sqrt(Im z)^2 <= |.z.| by A1,SQUARE_1:26;
  then -sqrt(Im z)^2 >= -|.z.| by XREAL_1:24;
  then Im z >= -|.Im z.| & -|.Im z.| >= -|.z.| by ABSVALUE:4,COMPLEX1:72;
  hence thesis by XXREAL_0:2;
end;
