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,n st UnionKers f = ker (f|^n) holds V is_the_direct_sum_of ker (f|^n),im (f
  |^n)
proof
  let V be finite-dimensional VectSp of K,f be linear-transformation of V,V,n;
  set KER=ker (f|^n);
  set IM=im (f|^n);
A1: dim V = rank (f|^n)+nullity (f|^n) by RANKNULL:44
    .= dim (IM+KER)+dim (IM/\KER) by VECTSP_9:32;
  assume
A2: UnionKers f = ker (f|^n);
  then (Omega).(IM/\KER) = (0).V by Th34
    .= (0).(IM/\KER) by VECTSP_4:36;
  then dim (IM/\KER)=0 by VECTSP_9:29;
  then (Omega).V=(Omega).(IM+KER) by A1,VECTSP_9:28;
  then
A3: KER + IM = (Omega).V by VECTSP_5:5;
  KER/\IM=(0).V by A2,Th34;
  hence thesis by A3,VECTSP_5:def 4;
end;
