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 Th14:
  f is with_eigenvalues & L is eigenvalue of f iff ker (f+(-L)*id
  V1) is non trivial
proof
  hereby
    assume f is with_eigenvalues & L is eigenvalue of f;
    then consider v1 such that
A1: v1<>0.V1 and
A2: f.v1=L*v1 by Def2;
    (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;
    then v1 in ker (f+(-L)*id V1) by RANKNULL:10;
    then
A3: v1 is Element of ker (f+(-L)*id V1);
    v1<>0.ker (f+(-L)*id V1) by A1,VECTSP_4:11;
    hence ker (f+(-L)*id V1) is non trivial by A3;
  end;
  assume ker (f+(-L)*id V1) is non trivial;
  then consider v be Vector of ker (f+(-L)*id V1) such that
A4: v<>0.ker (f+(-L)*id V1);
A5: v in ker (f+(-L)*id V1);
  reconsider v as Vector of V1 by VECTSP_4:10;
  0.V1 = (f+(-L)*id V1).v by A5,RANKNULL:10
    .= f.v+((-L)*id V1).v by MATRLIN:def 3
    .= f.v+(-L)*(id V1.v) by MATRLIN:def 4
    .= f.v+(-L)*v;
  then
A6: f.v = -(-L)*v by VECTSP_1:16
    .= (--L)*v by VECTSP_1:21
    .= L*v by RLVECT_1:17;
A7: v<>0.V1 by A4,VECTSP_4:11;
  then f is with_eigenvalues by A6;
  hence thesis by A6,A7,Def2;
end;
