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 Th31:
  f|(UnionKers (f+L*id V1)) is linear-transformation of UnionKers
  (f+L*id V1), UnionKers (f+L*id V1)
proof
  set fid= f+L*id V1;
  set U=UnionKers fid;
  reconsider fidU=fid|U as linear-transformation of U,U by Th30;
  rng (f|U) c= the carrier of U
  proof
    let y be object;
    assume y in rng (f|U);
    then consider x being object such that
A1: x in dom (f|U) and
A2: (f|U).x=y by FUNCT_1:def 3;
A3: x in the carrier of U by A1,FUNCT_2:def 1;
    then
A4: x in U;
    then x in V1 by VECTSP_4:9;
    then reconsider v=x as Vector of V1;
A5: (f|U).v =f.v by A1,FUNCT_1:47;
    dom fidU=the carrier of U by FUNCT_2:def 1;
    then fidU.v=fid.v & fidU/.v=fidU.v by A3,FUNCT_1:47,PARTFUN1:def 6;
    then
A6: fid.v in U;
    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 U by A4,VECTSP_4:21;
    then f.v in U by A7,A6,VECTSP_4:20;
    hence thesis by A2,A5;
  end;
  hence thesis by Lm1,FUNCT_2:6;
end;
