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 Th25:
  ker (f|^i) is Subspace of UnionKers f
proof
  the carrier of ker (f|^i) c= the carrier of UnionKers f
  proof
    let x be object;
    assume x in the carrier of ker (f|^i);
    then reconsider v=x as Element of ker (f|^i);
    (f|^i).v=0.V1 & v is Vector of V1 by RANKNULL:14,VECTSP_4:10;
    then x in UnionKers f by Th24;
    hence thesis;
  end;
  hence thesis by VECTSP_4:27;
end;
