reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;
reserve V for non trivial VectSp of K,
  V1,V2 for VectSp of K,
  f for linear-transformation of V1,V1,
  v,w for Vector of V,
  v1 for Vector of V1,
  L for Scalar of K;
reserve S for 1-sorted,
  F for Function of S,S;

theorem Th38:
  n >= 1 implies ex h be linear-transformation of V1,V1 st (f + L*
id V1) |^ n = f * h + ((L*id V1) |^ n) & for i holds (f |^ i) * h = h * (f |^ i
  )
proof
  set g=L*id V1;
  defpred P[Nat] means ex h be linear-transformation of V1,V1 st (f+g)|^$1 = f
  * h + (g|^$1) & for i holds (f |^i) * h = h * (f|^i);
A1: for n st 1<=n holds P[n] implies P[n+1]
  proof
    let n such that
    1<=n;
    assume P[n];
    then consider h be linear-transformation of V1,V1 such that
A2: (f+g)|^n = f * h + (g|^n) and
A3: for i holds (f |^i) * h = h * (f|^i);
    take H=f*h+(g|^n)+L*h;
A4: rng (f * h) c= [#] V1 by RELAT_1:def 19;
    thus (f+g)|^(n+1) = ((f+g)|^1)*((f+g)|^n) by Th20
      .= (f+g)*(f * h + (g|^n)) by A2,Th19
      .= (f*(f * h + (g|^n)))+ (g*(f * h + (g|^n))) by Lm6
      .= (f*(f * h + (g|^n)))+ (g*(f * h) + (g*(g|^n))) by Lm7
      .= (f*(f * h + (g|^n)))+ g*(f * h) + (g*(g|^n)) by Lm8
      .= (f*(f * h + (g|^n)))+ L*(id V1*(f * h))+(g*(g|^n)) by Lm4
      .= (f*(f * h + (g|^n)))+ L*(f * h) + (g*(g|^n)) by A4,RELAT_1:53
      .= (f*(f * h + (g|^n)))+ f * (L*h) + (g*(g|^n)) by Lm5
      .= f*H + (g*(g|^n)) by Lm7
      .= f*H + ((g|^1)*(g|^n)) by Th19
      .= f*H + (g|^(n+1)) by Th20;
    let i;
A5: (f |^i) *(f*h) = (f |^i) *f*h by RELAT_1:36
      .= ((f |^i) *(f|^1))*h by Th19
      .= (f |^(i+1))*h by Th20
      .= h*(f |^(i+1)) by A3
      .= h*((f |^1)* (f |^i)) by Th20
      .= (h*(f|^1))* (f |^i) by RELAT_1:36
      .= ((f|^1)*h)* (f |^i) by A3
      .= (f*h)* (f |^i) by Th19;
A6: (f |^i) * (g|^n) = (f |^i) * ((power K).(L,n)*id V1) by Lm9
      .= (power K).(L,n) *((f |^i) * id V1) by Lm5
      .= (power K).(L,n) *((f |^i) * (f|^0)) by Th18
      .= (power K).(L,n) *(f |^(i+0)) by Th20
      .= (power K).(L,n) *((f |^0) * (f|^i)) by Th20
      .= (power K).(L,n) *((id V1) * (f|^i)) by Th18
      .= ((power K).(L,n) *id V1) * (f|^i) by Lm4
      .= (g|^n) * (f|^i) by Lm9;
    (f |^i)*(L*h) = L*((f |^i)*h) by Lm5
      .= L*(h*(f|^i)) by A3
      .= (L*h)*(f|^i) by Lm4;
    hence (f |^i) * H = ((f |^i) *(f*h+(g|^n)))+(L*h)*(f|^i) by Lm7
      .= ((f*h)* (f |^i))+ ((g|^n) * (f|^i))+(L*h)*(f|^i) by A5,A6,Lm7
      .= (((f*h)+(g|^n)) * (f|^i))+(L*h)*(f|^i) by Lm6
      .= H*(f|^i) by Lm6;
  end;
A7: P[1]
  proof
    take h=id V1;
    thus (f+g)|^1 = f+g by Th19
      .= (f|^(1+0))+g by Th19
      .= (f|^1)*(f|^0)+g by Th20
      .= (f|^1)*h+g by Th18
      .= f*h+g by Th19
      .= f*h+(g|^1) by Th19;
    let i;
    thus (f |^i) * h = (f |^i) * (f|^0) by Th18
      .= f|^(i+0) by Th20
      .= (f |^0) * (f|^i) by Th20
      .= h * (f|^i) by Th18;
  end;
  for n st 1<=n holds P[n] from NAT_1:sch 8(A7,A1);
  hence thesis;
end;
