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
  f|im ((f+L*id V1)|^n) is linear-transformation of im ((f+L*id V1)|^n),
  im ((f+L*id V1)|^n)
proof
  set fid =f+L*id V1;
  set IM=im(fid|^n);
  reconsider fidI=fid|IM as linear-transformation of IM,IM by Th32;
  rng (f|IM) c= the carrier of IM
  proof
    let y be object;
    assume y in rng (f|IM);
    then consider x being object such that
A1: x in dom (f|IM) and
A2: (f|IM).x=y by FUNCT_1:def 3;
A3: x in the carrier of IM by A1,FUNCT_2:def 1;
    then
A4: x in IM;
    then x in V1 by VECTSP_4:9;
    then reconsider v=x as Vector of V1;
A5: (f|IM).v =f.v by A1,FUNCT_1:47;
    dom fidI=the carrier of IM by FUNCT_2:def 1;
    then fidI.v=fid.v & fidI/.v=fidI.v by A3,FUNCT_1:47,PARTFUN1:def 6;
    then
A6: fid.v in IM;
    fid.v = f.v+(L*id V1).v by MATRLIN:def 3
      .= f.v +L*((id V1).v) by MATRLIN:def 4
      .= f.v +L*v;
    then
A7: fid.v + (-L)*v = f.v + (L*v+ (-L)*v) by RLVECT_1:def 3
      .= f.v+((L+(-L))*v) by VECTSP_1:def 15
      .= f.v+(0.K *v) by VECTSP_1:16
      .= f.v+0.V1 by VECTSP_1:14
      .= f.v by RLVECT_1:def 4;
    (-L)*v in IM by A4,VECTSP_4:21;
    then f.v in IM by A7,A6,VECTSP_4:20;
    hence thesis by A2,A5;
  end;
  hence thesis by Lm1,FUNCT_2:6;
end;
