Proof of Proposition 19.20

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

Inv[3, 0, 3][
          z_] = ( c[3][
              z] - (1/3)*c[1][z]*c[2][
                z] + (2/27)*c[1][z]^3 - (1/2)*
              D[ c[2][z], z] + (1/3)*c[1][z]*
              D[c[1][z], z] + (1/6)*
              D[ c[1][z], {z, 2}]);

alpha[z_] = c[1][z]/3;

bet[z_] = (1/4)*(c[2][z] - alpha'[z] - 2*alpha[z]^2) // 
          Expand;

differ[z_] = c[3][z] - (2*bet'[z] + 4*alpha[z]*bet[z]) // 
          Expand;

Expand[differ[z] - Inv[3, 0, 3][z]]