reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem
  Im z <= |.z.|
proof
  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
A2: |.Im z.| <= |.z.| by Lm28;
  Im z <= |.Im z.| by Lm29;
  hence thesis by A2,XXREAL_0:2;
end;
