reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,L for Element of K;

theorem Th14:
  for V be VectSp of K,f be linear-transformation of V,V holds f|
  ker (f|^n) is nilpotent linear-transformation of ker (f|^n),ker (f|^n)
proof
  let V be VectSp of K,f be linear-transformation of V,V;
  set KER = ker (f|^n);
  reconsider fK=f|KER as linear-transformation of KER,KER by VECTSP11:28;
  now
    let v be Vector of KER;
    reconsider v1=v as Vector of V by VECTSP_4:10;
A1: v1 in KER;
    thus (fK|^n).v = ((f|^n) |KER).v by VECTSP11:22
      .= (f|^n).v1 by FUNCT_1:49
      .= 0.V by A1,RANKNULL:10
      .= 0.KER by VECTSP_4:11;
  end;
  hence thesis by Def4;
end;
