Lemma 3.1

Solve[{y==t x+s(a x+b y),z==a x+b y},{y,z}]

Solve[(1-b s)u+(t u+s v)(t+a s)+v(a+b t)==0,{u}]

Collect[Expand[(x-u)^2+(t x+s z-t u-s v)^2+(z-v)^2-(u^2+(t u+s v)^2+v^2)],{z,x}]

Solve[(-2 u-2 (t^2) u-2 s t v) x+(1+(t^2)) (x^2)+(-2 k+2 s t x) z+(1+(s^2)) (z^2)==0,z]

Simplify[s(t u+s v)+v/.u->((a v+a (s^2) v+b t v+s t v)/(-1+b s-a s t-(t^2)))]

Hence,  k is -(((-1+b s) (1+(s^2)+(t^2)) v)/(1-b s+a s t+(t^2)))

Simplify[1-(2k)^(-2)((((-2 k+2 s t x))^2)-4 (1+(s^2)) (-2 u x-2 (t^2) u x-2 s t v x+(x^2)+(t^2) (x^2)))]

Series[-(Sqrt[1-X]),{X,0,4}]

Factor[2 k t s-2(s^2+1)(u+t(t u+s v))/.{u->((a v+a (s^2) v+b t v+s t v)/(-1+b s-a s t-(t^2))),k->-(((-1+b s) (1+(s^2)+(t^2)) v)/(1-b s+a s t+(t^2)))}]

C0:=-((2 (a+a (s^2)+b t+s t) (1+(s^2)+(t^2)) v)/(-1+b s-a s t-(t^2)))

D0:=s^2+t^2+1

k:=-(((-1+b s) (1+(s^2)+(t^2)) v)/(1-b s+a s t+(t^2)))

A1 is

Simplify[-(u+t(t u+s v))/k/.u->((a v+a (s^2) v+b t v+s t v)/(-1+b s-a s t-(t^2)))]

A2 is

Simplify[D0/(2 k(s^2+1))+C0^2/(8 k^3(s^2+1))]

Here,

Factor[((s^2+t^2+1)(1-b s)^2+(a s^2+t s+b t+a)^2)]

A3 is

Simplify[2 C0 D0/(8 k^3(s^2+1))+C0^3/(16 k^5(s^2+1))]

A4 is

Simplify[D0^2/(8 k^3(s^2+1))+3 C0^2 D0/(16 k^5(s^2+1))+5 C0^4/(128 k^7(s^2+1))]

Factor[(-5 (((a (1+(s^2))+(b+s) t))^4)-6 (((-1+b s))^2) (((a (1+(s^2))+(b+s) t))^2) (1+(s^2)+(t^2))-(((-1+b s))^4) (((1+(s^2)+(t^2)))^2))]

Therefore A4 is

(((1-b s+a s t+(t^2)))^3) (1+(a^2)-2 b s+(a^2) (s^2)+(b^2) (s^2)+2 a b t+2 a s t+(t^2)+(b^2) (t^2)) (1+5 (a^2)-2 b s+(s^2)+10 (a^2) (s^2)+(b^2) (s^2)-2 b (s^3)+5 (a^2) (s^4)+(b^2) (s^4)+10 a b t+10 a s t+10 a b (s^2) t+10 a (s^3) t+(t^2)+5 (b^2) (t^2)+8 b s (t^2)+5 (s^2) (t^2)+(b^2) (s^2) (t^2))/(8 (((1-b s))^7) (((1+(s^2)+(t^2)))^3) (v^3))

Expand[(1+5 (a^2)-2 b s+(s^2)+10 (a^2) (s^2)+(b^2) (s^2)-2 b (s^3)+5 (a^2) (s^4)+(b^2) (s^4)+10 a b t+10 a s t+10 a b (s^2) t+10 a (s^3) t+(t^2)+5 (b^2) (t^2)+8 b s (t^2)+5 (s^2) (t^2)+(b^2) (s^2) (t^2))-((s^2+t^2+1)(1-b s)^2+5(a s^2+t s+b t+a)^2)]

Proposition 3.3

A2:=CT/(1-b s)

A3:=(DT+s CT1 A2)/(1-b s)

Expand[A3]

A4:=(ET+s CT1 A3+s DT1 A2+s^2 c2 A2^2)/(1-b s)

Expand[A4]

Simplify[a s^2+t s+b t+a/.t->(1-b s)T-a s]

Simplify[(a^2+b^2)s^2+2(a t-b)s+(b^2+1)t^2+2 a b t+a^2+1/.t->(1-b s)T-a s]

Here  R[T]:=1+(a^2)+2 a b T+(1+(b^2)) (T^2)

Solve[{(s CT1 CT+(1-b s)DT)RT==2(a+(b+s)T)CT^2},s]

this denominator is W:=b S+C K because

Expand[b(DT RT-2(b T+a)CT^2)+CT KT/.{KT->(2(b^2+1)T+2 a b)CT-RT CT1}]

Simplify[(s^2+t^2+1)(1-b s)^2+5(a s^2+t s+b t+a)^2/.t->(1-b s)T-a s]

Collect[Cancel[RT^2(1-b s)^3(A4-A2^3(1+(s^2)+5 (((a+(b+s) T))^2)+(((a s+(-1+b s) T))^2))/RT^2)],s,Simplify]

s:=ST/WT

Z1:=RT^2WT^2(1-b s)^3(A4-A2^3(1+(s^2)+5 (((a+(b+s) T))^2)+(((a s+(-1+b s) T))^2))/RT^2)

Collect[Cancel[Z1],{ET,DT,DT1,CT},Simplify]

As for the coefficient of CT^3,

Expand[(-(ST^2) (1+(a^2)+2 a b T+(5+(b^2)) (T^2))-8 ST T (a+b T) WT-(1+5 (a^2)+10 a b T+(1+5 (b^2)) (T^2)) (WT^2))+(WT^2+ST^2+5(a WT+(b WT+ST )T)^2+(a ST-(WT-b ST)T)^2)]

WT:=b ST+CT KT

ST:=DT RT-2(b T+a)CT^2

ZT:=KT^2(RT ET-CT^3)+RT KT DT(DT1 RT-3(b^2+1)T DT)+DT^2RT(-a b(2KT+T KT1)-2(a^2+1)(b^2+1)CT+((a^2+1)c2+(b^2+1)c0)RT)+2RT CT((b T+a)(DT KT1 CT+DT KT CT1-DT1 KT CT)-b DT CT KT)+4CT^4(b T+a)(((a^2-1)c2+(b^2+1)c0)(b T+a)-(1/2)a c1 RT1+2a(c2-c0)-b c1)

Expand[{Cancel[Z1]-RT CT^2 ZT/.{KT->RT1 CT-RT CT1,KT1->2(b^2+1)CT-2c2 RT}}/.{RT->(b^2+1)T^2+2 a b T+a^2+1,RT1->2(b^2+1)T+2 a b,CT->c0 +c1 T+ c2 T^2,CT1->c1+2 c2 T}]

Lemma 3.4

LTD:=4 ET

LTD1:=3 ET1

LTC:=3 DT

LTC1:=2 DT1

LTR:=4(b T+a)CT

LTa:=2 c0+c1 T

LTb:=CT1

LTbTa:=2 CT

LTR1:=4 b CT+2(b T+a)CT1

LTK:=3 RT1 DT-2 RT DT1+2 CT(2 b CT-(b T+a)CT1)

LTK1:=6 (b^2+1)DT-2(d2+3 T d3)RT+4(b c1-2 a c2)CT

LTc0:=3 d0+d1 T

LTc1:=2 d1+2 d2 T

LTc2:=d2+3 d3 T

ST1:=DT1 RT+DT RT1-2 b CT^2-4(b T+a)CT CT1

WT1:=b ST1+CT1 KT+CT KT1

LTS:=RT LTD+DT LTR-2 LTbTa CT^2-4(b T+a)CT LTC

LTW:=ST LTb+b LTS+KT LTC+CT LTK

Expand[{KT(WT LTS-ST LTW)+2 ST(WT ST1-ST WT1)-4 CT ZT/.{KT->RT1 CT-RT CT1,KT1->2(b^2+1)CT-2c2 RT}}/.{RT->(b^2+1)T^2+2 a b T+a^2+1,RT1->2(b^2+1)T+2 a b,CT->c0 +c1 T+ c2 T^2,CT1->c1+2 c2 T}]

Proposition 3.6.

(i)

Simplify[Expand[ZT/(c0-c2)/.{KT->2 T(c0-c2),RT->T^2+1,CT->c0+c2 T^2,a->0,b->0}]]

Collect[Expand[2 DT DT1 T (((1+(T^2)))^2)-(DT^2) (1+6 (T^2)+5 (T^4))-4 (c0-c2) (T^2) ((c0^3)+3 (c0^2) c2 (T^2)+3 c0 (c2^2) (T^4)+(c2^3) (T^6)-ET (1+(T^2)))/.{DT->d0+d1 T+d2 T^2+d3 T^3, DT1->d1+2 d2 T+3 d3 T^2,ET->e0+e1 T+e2 T^2+e3 T^3+e4 T^4}],T]

(iv)

a0:=s3 g7

a1:=-(s2 g7+s3 g6)

a2:=s1 g7+s2 g6+s3 g5

a3:=-(g7+s1 g6+s2 g5+s3 g4)

a4:=g6+s1 g5+s2 g4+s3 g3

a5:=-(g5+s1 g4+s2 g3+s3 g2)

a6:=g4+s1 g3+s2 g2+s3 g1

a7:=-(g3+s1 g2+s2 g1+s3)

a8:=g2+s1 g1+s2

a9:=-(g1+s1)

Solve[{a1==0,a9==0,a5-a3-a7==0},{s1,s2,s3}]

G1:=g1 g6-g3 g6+g5 g6-g7+g2 g7-g4 g7

G2:=g1 g2-g3-g1 g4+g5+g1 g6-g7

s1:=-g1

s2:=((g6 G2)/G1)

s3:=((-g7 G2)/G1)

a0

Factor[a0-a2+a4-a6+a8-1]

Simplify[(1+(g1^2)-2 g2+(g2^2)-2 g1 g3+(g3^2)+2 g4-2 g2 g4+(g4^2)+2 g1 g5-2 g3 g5+(g5^2)-2 g6+2 g2 g6-2 g4 g6+(g6^2)-2 g1 g7+2 g3 g7-2 g5 g7+(g7^2))-((((g1-g3+g5-g7))^2)+(((1-g2+g4-g6))^2))]

G3:=-(g1 g6-g7)

Collect[((((S1/3)+Y))^3)-S1 ((((S1/3)+Y))^2)+S2 ((S1/3)+Y)-S3,Y]

Factor[4(-(-((s1^2)/3)+s2)/3)^3-(-((2 (s1^3))/27)+((s1 s2)/3)-s3)^2]

G4:=-(g1^4) g2 (g6^3)+4 (g1^2) (g2^2) (g6^3)+(g1^3) g3 (g6^3)-8 g1 g2 g3 (g6^3)+(g1^3) g2 g3 (g6^3)+4 (g3^2) (g6^3)-(g1^2) (g3^2) (g6^3)+(g1^4) g4 (g6^3)-8 (g1^2) g2 g4 (g6^3)+8 g1 g3 g4 (g6^3)-(g1^3) g3 g4 (g6^3)+4 (g1^2) (g4^2) (g6^3)-(g1^3) g5 (g6^3)+8 g1 g2 g5 (g6^3)-(g1^3) g2 g5 (g6^3)-8 g3 g5 (g6^3)+2 (g1^2) g3 g5 (g6^3)-8 g1 g4 g5 (g6^3)+(g1^3) g4 g5 (g6^3)+4 (g5^2) (g6^3)-(g1^2) (g5^2) (g6^3)-(g1^4) (g6^4)+8 (g1^2) g2 (g6^4)-8 g1 g3 (g6^4)+(g1^3) g3 (g6^4)-8 (g1^2) g4 (g6^4)+8 g1 g5 (g6^4)-(g1^3) g5 (g6^4)+4 (g1^2) (g6^5)+4 (g1^5) (g6^2) g7-17 (g1^3) g2 (g6^2) g7-(g1^3) (g2^2) (g6^2) g7+17 (g1^2) g3 (g6^2) g7-8 (g1^4) g3 (g6^2) g7+19 (g1^2) g2 g3 (g6^2) g7-18 g1 (g3^2) (g6^2) g7+4 (g1^3) (g3^2) (g6^2) g7+17 (g1^3) g4 (g6^2) g7+2 (g1^3) g2 g4 (g6^2) g7-19 (g1^2) g3 g4 (g6^2) g7-(g1^3) (g4^2) (g6^2) g7-17 (g1^2) g5 (g6^2) g7+8 (g1^4) g5 (g6^2) g7-19 (g1^2) g2 g5 (g6^2) g7+36 g1 g3 g5 (g6^2) g7-8 (g1^3) g3 g5 (g6^2) g7+19 (g1^2) g4 g5 (g6^2) g7-18 g1 (g5^2) (g6^2) g7+4 (g1^3) (g5^2) (g6^2) g7-16 (g1^3) (g6^3) g7-8 g1 g2 (g6^3) g7-(g1^3) g2 (g6^3) g7+8 g3 (g6^3) g7+17 (g1^2) g3 (g6^3) g7+8 g1 g4 (g6^3) g7+(g1^3) g4 (g6^3) g7-8 g5 (g6^3) g7-17 (g1^2) g5 (g6^3) g7-8 g1 (g6^4) g7-8 (g1^4) g6 (g7^2)+45 (g1^2) g2 g6 (g7^2)+8 (g1^4) g2 g6 (g7^2)-18 (g1^2) (g2^2) g6 (g7^2)-45 g1 g3 g6 (g7^2)+8 (g1^3) g3 g6 (g7^2)-9 g1 g2 g3 g6 (g7^2)-8 (g1^3) g2 g3 g6 (g7^2)+27 (g3^2) g6 (g7^2)-45 (g1^2) g4 g6 (g7^2)-8 (g1^4) g4 g6 (g7^2)+36 (g1^2) g2 g4 g6 (g7^2)+9 g1 g3 g4 g6 (g7^2)+8 (g1^3) g3 g4 g6 (g7^2)-18 (g1^2) (g4^2) g6 (g7^2)+45 g1 g5 g6 (g7^2)-8 (g1^3) g5 g6 (g7^2)+9 g1 g2 g5 g6 (g7^2)+8 (g1^3) g2 g5 g6 (g7^2)-54 g3 g5 g6 (g7^2)-9 g1 g4 g5 g6 (g7^2)-8 (g1^3) g4 g5 g6 (g7^2)+27 (g5^2) g6 (g7^2)+62 (g1^2) (g6^2) (g7^2)-17 (g1^2) g2 (g6^2) (g7^2)-45 g1 g3 (g6^2) (g7^2)+17 (g1^2) g4 (g6^2) (g7^2)+45 g1 g5 (g6^2) (g7^2)+4 (g6^3) (g7^2)+4 (g1^3) (g7^3)-27 g1 g2 (g7^3)-8 (g1^3) g2 (g7^3)+27 g1 (g2^2) (g7^3)+4 (g1^3) (g2^2) (g7^3)+27 g3 (g7^3)-27 g2 g3 (g7^3)+27 g1 g4 (g7^3)+8 (g1^3) g4 (g7^3)-54 g1 g2 g4 (g7^3)-8 (g1^3) g2 g4 (g7^3)+27 g3 g4 (g7^3)+27 g1 (g4^2) (g7^3)+4 (g1^3) (g4^2) (g7^3)-27 g5 (g7^3)+27 g2 g5 (g7^3)-27 g4 g5 (g7^3)-72 g1 g6 (g7^3)+45 g1 g2 g6 (g7^3)+27 g3 g6 (g7^3)-45 g1 g4 g6 (g7^3)-27 g5 g6 (g7^3)+27 (g7^4)-27 g2 (g7^4)+27 g4 (g7^4)

g1:=SymmetricPolynomial[1,{x1,x2,x3,x4,x5,x6,x7}]

g2:=SymmetricPolynomial[2,{x1,x2,x3,x4,x5,x6,x7}]

g3:=SymmetricPolynomial[3,{x1,x2,x3,x4,x5,x6,x7}]

g4:=SymmetricPolynomial[4,{x1,x2,x3,x4,x5,x6,x7}]

g5:=SymmetricPolynomial[5,{x1,x2,x3,x4,x5,x6,x7}]

g6:=SymmetricPolynomial[6,{x1,x2,x3,x4,x5,x6,x7}]

g7:=SymmetricPolynomial[7,{x1,x2,x3,x4,x5,x6,x7}]

G1/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

G2/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

G3/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

G4/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a0/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a1/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a2/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a3/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a4/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a5/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a6/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a7/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a8/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a9/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a5-a3-a7/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

a0-a2+a4-a6+a8-1/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}

NSolve[(a0+a1 X+a2 X^2+a3 X^3+a4 X^4+a5 X^5+a6 X^6+a7 X^7+a8 X^8+a9 X^9+X^{10}==0/.{x1->1,x2->-2,x3->2,x4->3,x5->4,x6->-10,x7->1/10}),{X}]

Lemma 3.8

Solve[y==a x +b(x^2+y^2),y]

Series[((1-(Sqrt[1-4 a b x-4 (b^2) (x^2)]))/(2 b)),{x,0,4}]

P[x]:=c0 x^2+c2 y^2+d0 x^3+d1 x^2y+d2 x y^2+d3 y^3+e0 x^4+e1 x^3 y+e2 x^2 y^2+e3 x y^3+e4 y^4/.{y->a x+(1+a^2)b x^2+2 a b^2(1+a^2)x^3+(1+a^2)(1+5 a^2)b^3 x^4}

Coefficient[P[x],x,2]

Coefficient[P[x],x,3]

Coefficient[P[x],x,4]

The proof of Theorem 2.5.

To derive the explicit form of NT, we identify ZT with a function ZT[u], or ZT[T] where u and T are variables for differentiation   in L_T and in T,respectively. (u is a fictional variable for L_T).

Therefore, LTZ:L_T[ZT] is given by

ZTu[u]:=ZT/.{KT->KT[u],RT->RT[u],ET->ET[u],CT->CT[u],DT->DT[u],DT1->DT1[u],b->b[u],a->a[u],KT1->KT1[u],c2->c2[u],c0->c0[u],CT1->CT1[u],c1->c1[u],RT1->RT1[u]}

LTZ:=D[ZTu[u],u]/.{KT'[u]->LTK,KT[u]->KT,RT'[u]->LTR,RT[u]->RT,ET'[u]->LTE,ET[u]->ET,CT'[u]->LTC,CT[u]->CT,DT'[u]->LTD,DT[u]->DT,DT1'[u]->LTD1,DT1[u]->DT1,b'[u]->LTb, b[u]->b,a'[u]->LTa, a[u]->a,KT1'[u]->LTK1,KT1[u]->KT1,c2'[u]->LTc2,c2[u]->c2,c0'[u]->LTc0,c0[u]->c0,CT1'[u]->LTC1,CT1[u]->CT1,c1'[u]->LTc1,c1[u]->c1,RT1'[u]->LTR1,RT1[u]->RT1}

Z'[T] is given by the following ZT1:

ZTT[T]:=ZT/.{KT->KT[T],RT->RT[T],ET->ET[T],CT->CT[T],DT->DT[T],DT1->DT1[T],KT1->KT1[T],CT1->CT1[T],RT1->RT1[T]}

ZT1:=D[ZTT[T],T]/.{KT'[T]->KT1,KT[T]->KT,RT'[T]->RT1,RT[T]->RT,ET'[T]->ET1,ET[T]->ET,CT'[T]->CT1,CT[T]->CT,DT'[T]->DT1,DT[T]->DT,DT1'[T]->DT2,DT1[T]->DT1,KT1'[T]->KT2,KT1[T]->KT1,CT1'[T]->2 c2,CT1[T]->CT1,RT1'[T]->RT2,RT1[T]->RT1}

The explicit form of N[T] is the following:

NT:=-5RT KT^2ET1(RT DT-2(b T+a)CT^2)+DT^3RT B1+2DT^2DT1 RT^2B2-DT^2RT^2KT ((3d3 T+d2)(5RT-(b^2+1)T^2)+(d1 T+3d0)(b^2+1))+DT^2B3+2(b T+a)DT DT1 RT CT B4+10(b T+a)DT DT2 RT^2KT CT^2+DT B5-4(b T+a)DT1 RT KT CT^3(5(b T+a) CT1+2b CT)+4(b T+a)CT^4KT(3d0 B6+d1 B7+d2 B8-3d3 T B9)+4CT^4B10

B1:=KT(42(b^2+1)RT-20(a^2+b^2+1))-4c2 RT(3a b RT-2(a^2+b^2+1-a^2b^2)T+2a b(a^2+1))+2(b^2+1)c1 RT(3RT+4a b T+4a^2+4)-4(b^2+1)RT RT1 c0

B2:=-(4(b^2+1)T+5a b)KT-((b^2+1)T^2+a^2+1)KT1+4(a^2+1)(b^2+1)CT-2((a^2+1)c2+(b^2+1)c0)RT

B3:=KT^2(-4a RT CT1+12(b^2+1)(b T+a)T CT+8a b(b T+a)CT+12b RT CT+2b RT(2c0+c1 T))+KT KT1(4a b T(b T+a)CT-12(b T+a)RT CT+b T RT(2c0+c1 T)+a T RT CT1)+KT(-4a(b^2+6)RT CT^2-(18(b^2+1)T(b T+a)+16a b(b T+a)-4b(a^2+1))RT CT CT1+8(a^2+1)(b^2+1)(b T+a)CT^2-8(b T+a)((a^2+1)c2+(b^2+1)c0)RT CT+4a b T(b c1-2a c2)RT CT+4a(b^2+1)(2c0+c1 T)RT CT-2a c2(2c0+c1 T)RT^2-2b c0 CT1 RT^2-12(b^2+1)(b T+a)T RT1 CT^2-8a b(b T+a)RT1 CT^2-4(b T+a)RT RT1 CT CT1+4b RT RT1 CT^2-4(b T+a)RT^2CT1^2-8c2(b T+a)RT^2CT+4b RT^2CT CT1)-2(b T+a)KT1 CT(2a b T RT1 CT+5RT1 RT CT+6RT^2CT1)-4(b T+a)KT2 RT CT^2(RT+a b T)-8(a^2+1)(b^2+1)(b T+a)RT1 CT^3+8(b T+a)((a^2+1)c2+(b^2+1)c0)RT RT1 CT^2-8(a^2+1)(b^2+1)(b T+a)RT CT^2CT1

B4:=2(3(b^2+1)T+5a b)KT CT+5RT KT CT1+2((b^2+1)T^2+a^2+1)KT1 CT-8(a^2+1)(b^2+1)CT^2+4RT CT((a^2+1)c2+(b^2+1)c0)

B5:=9CT^2KT^3+KT^2CT^2(6RT1 CT+4RT CT1+8b(b T+a)CT-8(b T+a)^2CT1)-8(b T+a)^2KT KT1 CT^3+KT CT^2(-8(b T+a)(b c1-2a c2)RT CT-8b(b T+a)RT CT1 CT+8b^2RT CT^2-4b(b T+a)RT CT CT1-48(b T+a)^2CT((a^2-1)c2+(b^2+1)c0)+24a c1(b T+a)RT1 CT-48(b T+a)CT(2a(c2-c0)-b c1)+12(b T+a)^2RT CT1^2-8b(b T+a)RT1 CT^2+8(b T+a)^2RT1 CT CT1+16c2(b T+a)^2RT CT-8b(b T+a)RT CT CT1)+8(b T+a)^2KT1 CT^3(RT1 CT+3RT CT1)+8(b T+a)^2RT CT^4KT2-16b(b T+a)RT CT^4((a^2-1)c2+(b^2+1)c0)-32(b T+a)^2RT CT^3CT1((a^2-1)c2+(b^2+1)c0)+16a(b T+a)RT RT1 c1 CT^3CT1+4a b RT RT1 c1 CT^4+8a(b^2+1)(b T+a)RT c1 CT^4-8b RT CT^4(2a(c2-c0)-b c1)-32(b T+a)RT CT^3CT1(2a(c2-c0)-b c1)

B6:=-(b^2+1)(b T+a)+2a

B7:=-(b^2+1)T(b T+a)+a RT1+2(a T+b)

B8:=-(a^2-3)(b T+a)+a T RT1-4a-4(b T+a)RT

B9:=(a^2-1)(b T+a)+2a+4(b T+a)RT

B10:=KT^2(2b CT-4(b T+a)CT1)-2(b T+a)CT KT KT1+KT(a RT1 c1 CT-4(b T+a)CT((a^2-1)c2+(b^2+1)c0)-4a(c2-c0)CT+2b c1 CT-2a(b T+a)^2c2(2c0+c1 T)-2b(b T+a)^2c0 CT1+2a b(b T+a)c1 CT+a(b T+a)^2c1 CT1+(1/2)(b T+a)RT1 c1 (2c0+c1 T)-2(b T+a)(c2-c0)(2c0+c1 T)+(b T+a)c1 CT1)+8b(b T+a)^2((a^2-1)c2+(b^2+1)c0)CT^2+16(b T+a)^3CT CT1((a^2-1)c2+(b^2+1)c0)-8a(b T+a)^2RT1 c1 CT CT1-2a b(b T+a)RT1 c1 CT^2-4a(b^2+1)(b T+a)^2c1 CT^2+4b(b T+a)CT^2(2a(c2-c0)-b c1)+16(b T+a)^2CT CT1(2a(c2-c0)-b c1)

On the other hand, the definition of N[T] is given by the following:

NTdef:=-KT(LTZ-KT^2 RT LTE)-2 ST ZT1

Expand[{NT-NTdef/.{KT->RT1 CT-RT CT1,KT1->2(b^2+1)CT-2c2 RT,KT2->2(b^2+1)CT1-2 c2 RT1}}/.{RT->(b^2+1)T^2+2 a b T+a^2+1,RT1->2(b^2+1)T+2 a b,RT2->2(b^2+1),CT->c0 +c1 T+ c2 T^2,CT1->c1+2 c2 T,CT2->2 c2,DT->d0+d1 T+d2 T^2+d3 T^3,DT1->d1+2 T d2+3 d3 T^2,DT2->2 d2 +6 d3 T}]

Simplify[Coefficient[{NT/.{KT->RT1 CT-RT CT1,KT1->2(b^2+1)CT-2c2 RT,KT2->2(b^2+1)CT1-2 c2 RT1}}/.{RT->(b^2+1)T^2+2 a b T+a^2+1,RT1->2(b^2+1)T+2 a b,RT2->2(b^2+1),CT->c0 +c1 T+ c2 T^2,CT1->c1+2 c2 T,CT2->2 c2,DT->d0+d1 T+d2 T^2+d3 T^3,DT1->d1+2 T d2+3 d3 T^2,DT2->2 d2 +6 d3 T,ET1->e1+2 e2 T+3 e3 T^2+4 e4 T^3},T,14]]