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
  for K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V1 be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty
  ModuleStr over K for W be Subspace of V1, f be Function of V1,V1, fW be
  Function of W,W st fW=f|W holds (f|^n)|W=fW|^n
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V1 be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty
  ModuleStr over K;
  let W be Subspace of V1,f be Function of V1,V1,fW be Function of W,W such
  that
A1: fW=f|W;
  defpred P[Nat] means (f|^$1)|W=fW|^$1;
A2: for n st P[n] holds P[n+1]
  proof
    let n such that
A3: P[n];
A4: rng (fW|^n) c= [#]W by RELAT_1:def 19;
    thus (f|^(n+1))|W = ((f|^1)*(f|^n))|W by Th20
      .= (f|^1)*((f|^n)|W) by RELAT_1:83
      .= (f|^1)*((id W)*(fW|^n)) by A3,A4,RELAT_1:53
      .= ((f|^1)*(id W))*(fW|^n) by RELAT_1:36
      .=(f*id W)*(fW|^n) by Th19
      .= fW*(fW|^n) by A1,RELAT_1:65
      .= (fW|^1)*(fW|^n) by Th19
      .=fW|^(n+1) by Th20;
  end;
  [#]W c= [#]V1 by VECTSP_4:def 2;
  then
A5: [#]W=[#]V1/\[#]W by XBOOLE_1:28;
  (f|^0)|W = (id V1)|W by Th18
    .= (id V1)*id W by RELAT_1:65
    .= id W by A5,FUNCT_1:22
    .= fW|^0 by Th18;
  then
A6: P[0];
  for n holds P[n] from NAT_1:sch 2(A6,A2);
  hence thesis;
end;
