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 V be finite-dimensional VectSp of K,f be linear-transformation of
  V,V ex n st UnionKers f = ker (f|^n)
proof
  let V be finite-dimensional VectSp of K;
  let f be linear-transformation of V,V;
  defpred P[Nat] means for n holds dim ker(f|^n)<=$1;
  P[dim UnionKers f]
  proof
    let n;
    ker (f|^n) is Subspace of UnionKers f by Th25;
    hence thesis by VECTSP_9:25;
  end;
  then
A1: ex k st P[k];
  ex k st P[k] & for n st P[n] holds k<=n from NAT_1:sch 5(A1);
  then consider k such that
A2: P[k] and
A3: for n st P[n] holds k<=n;
  ex m st dim ker(f|^m)=k
  proof
    assume
A4: for m holds dim ker(f|^m)<>k;
    dim ker (f|^0)<=k by A2;
    then dim ker(f|^0)<k by A4,XXREAL_0:1;
    then reconsider k1=k-1 as Element of NAT by NAT_1:20;
    now
      let n;
      dim ker (f|^n)<=k by A2;
      then dim ker(f|^n)<k1+1 by A4,XXREAL_0:1;
      hence dim ker(f|^n)<=k1 by NAT_1:13;
    end;
    then k1+1<=k1 by A3;
    hence thesis by NAT_1:16;
  end;
  then consider m such that
A5: dim ker(f|^m)=k;
  take m;
  assume
A6: UnionKers f<>ker (f|^m);
  ker (f|^m) is Subspace of UnionKers f by Th25;
  then consider v be Element of UnionKers f such that
A7: not v in ker (f|^m) by A6,VECTSP_4:32;
A8: v in UnionKers f;
  reconsider v as Vector of V by VECTSP_4:10;
  consider i such that
A9: (f|^i).v=0.V by A8,Th24;
  i>m
  proof
    assume i<=m;
    then reconsider j=m-i as Element of NAT by NAT_1:21;
    (f|^(j+i)).v=0.V by A9,Th23;
    hence thesis by A7,RANKNULL:10;
  end;
  then reconsider j=i-m as Element of NAT by NAT_1:21;
A10: ker (f|^m) is Subspace of ker (f|^(m+j)) by Th26;
  then
A11: k<=dim ker(f|^i) by A5,VECTSP_9:25;
  (Omega).(ker (f|^m))<>(Omega).(ker(f|^i)) by A7,A9,RANKNULL:10;
  then k <> dim ker (f|^i) by A5,A10,VECTSP_9:28;
  then k < dim ker (f|^i) by A11,XXREAL_0:1;
  hence thesis by A2;
end;
