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

theorem Th17:
  for K be Field, V be VectSp of K, W be Subspace of V for f be
  nilpotent Function of V,V st f|W is Function of W,W holds f|W is nilpotent
  Function of W,W
proof
  let K be Field,V be VectSp of K,W be Subspace of V;
  let f be nilpotent Function of V,V;
  assume f|W is Function of W,W;
  then reconsider fW=f|W as Function of W,W;
  consider n such that
A1: f|^n =ZeroMap(V,V) by Th13;
  [#]W c= [#]V by VECTSP_4:def 2;
  then
A2: [#]W=[#]V/\[#]W by XBOOLE_1:28;
  fW|^n = ZeroMap(V,V) |W by A1,VECTSP11:22
    .= ((the carrier of V)-->0.V) |[#]W by GRCAT_1:def 7
    .= ((the carrier of V)/\[#]W) -->0.V by FUNCOP_1:12
    .= (the carrier of W)-->0.W by A2,VECTSP_4:11
    .= ZeroMap(W,W) by GRCAT_1:def 7;
  hence thesis by Th13;
end;
