The correctness of the formulas (1.81) and (1.82) was checked with the Mathematica notebook for page 16.  Here, we evaluate the Input commands for Section 16.2 that show how the algorithmic proof for Theorem 14.3  enabled the right members of (1.81) and (1.82) to be discovered.

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

B[1, 1][z_] = w[1, 1][z] - w[0, 1][z]^2

B[1, 2][
        z_] = w[1, 2][z] - w[0, 1][z]*w[0, 2][z]

B[2, 2][z] = w[2, 2][z] - w[0, 2][z]^2

x[2, 2][
          z_] = (  Inv[m, 2, 2][
              z] + (2 (m - 1)/m)*w[0, 1][z]*w[1, 2][
                z] - (6 (m - 1)/(5 (m + 1) m)) w[0, 2][
                z]*w[1, 1][z] + w[0, 2][
                z]^2 - ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) (w[0, 
                      1][z])^2*w[1, 1][
                z] - (2 (5  m + 
                      2) (m - 1)/(5 (m + 1) m)) (w[0, 
                      1][z])^2*w[0, 2][
                z] + ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m^2)) (w[0, 
                      1][z])^4 + ((m - 1)/3)*
              D[w[1, 2][z], z] - ((m - 1)^2/30)*
              D[w[1, 1][z], {z, 2}] - ((m - 1)^2/(3  m))*w[
                  0, 1][z]*
              D[w[1, 1][z], 
                z] - ((5  m - 
                      4) (m - 1)^2/(15 (m + 1) m))*
              D[w[0, 1][z], z]*w[1, 1][
                z] - ((m - 1)/3)*w[0, 1][z]*
              D[w[0, 2][z], z] - ((m - 1)/3)*
              D[w[0, 1][z], z]*w[0, 2][
                z] + ((m - 1)^2/15)*w[0, 1][z]*
              D[w[0, 1][z], {z, 2}] + ((m - 1)^2/
                  15)*(D[w[0, 1][z], 
                    z])^2 + ((5  m + 
                      2) (m - 1)^2/(5 (m + 1) m))*(w[0, 
                      1][z])^2*D[w[0, 1][z], z] );

x[1, 2][
          z_] = (Inv[m, 1, 2][
              z] + ((m - 1)/m)*w[0, 1][z]*w[1, 1][
                z] + w[0, 1][z]*w[0, 2][
                z] - ((m - 1)/
                  m) (w[0, 1][z])^3 + ((m - 1)/4)*
              D[w[1, 1][z], z] - ((m - 1)/2) w[0, 1][z]*
              D[w[0, 1][z], z]);

x[1, 1][
          z_] = Inv[m, 1, 1][z] + (w[0, 1][z])^2;

x[0, 2][z_] = ( 
          Binomial[m + 1, 3]*a[m, 0, 2][z] + ((m - 1)/2)*
              D[w[0, 1][z], 
                z] + ((m - 1)/(2  m)) (w[0, 1][
                    z])^2);

m  =  2

E6[
          z_] = (B[1, 1][z]*B[2, 2][z] - 
            2  B[1, 2][z]^2 + w[0, 1][z]*B[1, 1][z]*B[1, 2][
                z] - w[0, 2][z]*B[1, 1][z]^2 + B[1, 1][z]*
              D[B[1, 2][z], z] - 
            D[B[1, 1][z], z]*B[1, 2][z]);

H7[
          z_] = (B[1, 1][z]*D[B[2, 2][z], z] - 
            D[B[1, 1][z], z]*B[2, 2][z] - 
            2  B[1, 2][z]*B[2, 2][z] + 
            2 w[0, 1][z]*B[1, 1][z]*B[2, 2][z] - 
            2 w[0, 2][z]*B[1, 1][z]*B[1, 2][z]);

E7[z_] = 
        H7[z] - (1/6) (D[E6[z], z] + 
                6  w[0, 1][z]*E6[z]);

{B1, C1} = 
        Expand[{E6[z], 
              E7[z]} /. w[2, 2]'[z] -> x[2, 2]'[z] ];

{B2, C2} = 
        Expand[{B1, C1} /. w[2, 2][z] -> x[2, 2][z]];

{B3, C3} = 
        Expand[{B2, 
              C2} /. w[1, 2]''[z] -> x[1, 2]''[
                z]];

{B4, C4} = 
        Expand[{B3, C3} /. w[1, 2]'[z] -> x[1, 2]'[z]];

{B5, C5} = 
        Expand[{B4, C4} /. w[1, 2][z] -> x[1, 2][z]];

{B6, C6} = 
        Expand[{B5, 
              C5} /. w[1, 1]''[z] -> x[1, 1]''[
                z]];

{B7, C7} = 
        Expand[{B6, 
              C6} /. w[1, 1]'[z] -> x[1, 1]'[z] ];

{B8, C8} = 
        Expand[{B7, C7} /. w[1, 1][z] -> x[1, 1][z] ];

{B9, C9} = 
        Expand[{B8, C8} /. w[0, 2][z] -> x[0, 2][z] ];

B9

Coefficient[ 
      B9,  Inv[2, 1, 1][z]*Inv[2, 2, 2][z] ]

Coefficient[ B9,  Inv[2, 1, 2][z]^2 ]

Coefficient[ 
      B9,  Inv[2, 1, 1][z]*D[Inv[2, 1, 2][z], z] ]

Coefficient[ 
      B9,  Inv[2, 1, 2][z]*D[Inv[2, 1, 1][z], z] ]

Coefficient[ 
      B9,   a[2, 0, 2][z]*Inv[2, 1, 1][z]^2 ]

Coefficient[ 
      B9,  Inv[2, 1, 1][z]*D[Inv[2, 1, 1][z], {z, 2}] ]

Coefficient[ B9,  D[Inv[2, 1, 1][z],  z]^2 ]

By this means, we discovered the right member of (1.81).

C9

Coefficient[ C9, Inv[2, 1, 2][z]*Inv[2, 2, 2][z] ]

Coefficient[ 
      C9,  Inv[2, 1, 1][z]*D[Inv[2, 2, 2][z], z] ]

Coefficient[ 
      C9,  Inv[2, 2, 2][z]*D[Inv[2, 1, 1][z], z] ]

Coefficient[ 
      C9,  a[2, 0, 2][z]* Inv[2, 1, 1][z]*Inv[2, 1, 2][
          z]  ]

Coefficient[ C9,  
      D[ Inv[2, 1, 1][z],  {z, 2}]*Inv[2, 1, 2][z] ]

Coefficient[ C9,  
      D[ Inv[2, 1, 1][z],  z]*D[Inv[2, 1, 2][z], z] ]

Coefficient[ 
      C9,  Inv[2, 1, 1][z]*D[Inv[2, 1, 2][z], {z, 2}] ]

By this means, we discovered the right member of (1.82).

In order to check (1.81) and (1.82) again, we introduce

Inv[2, 1, 1][
          z_] = w[1, 1][z] - (w[0, 1][z])^2;

Inv[2, 1, 2][
          z_] = (w[1, 2][
              z] - (1/2) w[0, 1][z]*w[1, 1][z] - w[0, 1][
                z]*w[0, 2][
                z] + (1/2) (w[0, 1][z])^3 - (1/4) 
              D[w[1, 1][z], z] + (1/2) w[0, 1][z]*
              D[ w[0, 1][z], z]);

Inv[2, 2, 2][
          z_] = (w[2, 2][
              z] - w[0, 1][z]*w[1, 2][z] + (1/5) w[0, 2][
                z]*w[1, 1][z] - w[0, 2][
                z]^2 + (1/5) w[0, 1][z]^2*w[1, 1][
                z] + (4/5) (w[0, 1][z])^2*w[0, 2][
                z] - (1/5) (w[0, 1][z])^4 - (1/3) 
              D[w[1, 2][z], z] + (1/30) 
              D[w[1, 1][z],  {z, 2}] + (1/6) w[0, 1][z]*
              D[ w[1, 1][z], z] + (1/15) 
              D[w[0, 1][z], z]*w[1, 1][z] + (1/3) w[0, 1][z]*
              D[w[0, 2][z], z] + (1/3) 
              D[w[0, 1][z], z]*w[0, 2][z] - (1/15) w[0, 1][z]*
              D[w[0, 1][z], {z, 2}] - (1/15) 
              D[w[0, 1][z], z]^2 - (2/5) (w[0, 1][z])^2*
              D[w[0, 1][z], z]);

a[2, 0, 2][
          z_] = w[0, 2][z] - (1/2) 
            D[ w[0, 1][z],  
              z] - (1/4) (w[0, 1][z])^2;

and check

Expand[
      E6[z] - (Inv[2, 1, 1][z]*Inv[2, 2, 2][z] - 
            2*Inv[2, 1, 2][z]^2 + (4/
                  3) (Inv[2, 1, 1][z]*
                    D[Inv[2, 1, 2][z], z] - (3/2) Inv[2, 1, 2][
                      z]*D[Inv[2, 1, 1][z], z]) - (6/
                  5) ( a[2, 0, 2][
                      z]*Inv[2, 1, 1][z]^2 - (1/4) Inv[2, 1, 1][
                      z]*D[Inv[2, 1, 1][z], {z, 2}] + (5/
                        16) D[Inv[2, 1, 1][z], z]^2))]

as well as

Expand[
      E7[z] - (-2 Inv[2, 1, 2][z]*Inv[2, 2, 2][
                z] + (5/
                  6) ( Inv[2, 1, 1][z]*D[Inv[2, 2, 2][z], z] - 
                  2 Inv[2, 2, 2][z]*D[Inv[2, 1, 1][z], z]) + (1/
                  9) ( Inv[2, 1, 1][z]*
                    D[Inv[2, 1, 2][z], {z, 2}] - (7/
                        2) D[ Inv[2, 1, 1][z],  z]*
                    D[Inv[2, 1, 2][z], z] + (21/10) 
                    D[ Inv[2, 1, 1][z], {z, 2}]*Inv[2, 1, 2][
                      z] - (72/5) a[2, 0, 2][z]* Inv[2, 1, 1][
                      z]*Inv[2, 1, 2][
                      z]))]

This completes the deduction and check of (1.81) and (1.82).