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 Th26:
  ker (f|^i) is Subspace of ker (f|^(i+j))
proof
  the carrier of ker (f|^i) c= the carrier of ker (f|^(i+j))
  proof
    let x be object such that
A1: x in the carrier of ker(f|^i);
    reconsider v = x as Vector of V1 by A1,VECTSP_4:10;
    (f|^i).v = 0.V1 by A1,RANKNULL:14;
    then (f|^(i+j)).v = 0.V1 by Th23;
    then v in ker (f|^(i+j)) by RANKNULL:10;
    hence thesis;
  end;
  hence thesis by VECTSP_4:27;
end;
