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;

theorem
  ker f is non trivial implies f is with_eigenvalues & 0.K is eigenvalue of f
proof
  assume ker f is non trivial;
  then consider v be Vector of ker f such that
A1: v<>0.ker f;
  reconsider v as Vector of V1 by VECTSP_4:10;
A2: f.v = 0.V1 by RANKNULL:14
    .= 0.K*v by VECTSP_1:14;
A3: v<>0.V1 by A1,VECTSP_4:11;
  then f is with_eigenvalues by A2;
  hence thesis by A3,A2,Def2;
end;
