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 Th17:
  for f,L st f is with_eigenvalues & L is eigenvalue of f holds v1
  is eigenvector of f,L iff v1 in ker (f+(-L)*id V1)
proof
  let f,L such that
A1: f is with_eigenvalues & L is eigenvalue of f;
  hereby
    assume v1 is eigenvector of f,L;
    then
A2: f.v1=L*v1 by A1,Def3;
    (f+(-L)*id V1).v1 = f.v1+((-L)*id V1).v1 by MATRLIN:def 3
      .= f.v1+(-L)*(id V1.v1) by MATRLIN:def 4
      .= f.v1+(-L)*v1
      .= (L+-L)*v1 by A2,VECTSP_1:def 15
      .= 0.K*v1 by VECTSP_1:19
      .= 0.V1 by VECTSP_1:15;
    hence v1 in ker (f+(-L)*id V1) by RANKNULL:10;
  end;
  assume v1 in ker (f+(-L)*id V1);
  then 0.V1 = (f+(-L)*id V1).v1 by RANKNULL:10
    .= f.v1+((-L)*id V1).v1 by MATRLIN:def 3
    .= f.v1+(-L)*(id V1.v1) by MATRLIN:def 4
    .= f.v1+(-L)*v1;
  then f.v1 = -(-L)*v1 by VECTSP_1:16
    .= (--L)*v1 by VECTSP_1:21
    .= L*v1 by RLVECT_1:17;
  hence thesis by A1,Def3;
end;
