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
  for a st a <> 0.K for f be with_eigenvalues Function of V,V for L be
eigenvalue of f holds a*f is with_eigenvalues & a*L is eigenvalue of a*f & ( w
  is eigenvector of f,L iff w is eigenvector of a*f,a*L )
proof
  let a such that
A1: a<>0.K;
  let f be with_eigenvalues Function of V,V;
  let L be eigenvalue of f;
  consider v such that
A2: v <> 0.V and
A3: f.v = L * v by Def2;
A4: (a*f).v = a*(L*v) by A3,MATRLIN:def 4
    .= (a*L)*v by VECTSP_1:def 16;
  hence
A5: a*f is with_eigenvalues by A2;
  hence
A6: a*L is eigenvalue of a*f by A2,A4,Def2;
  hereby
    assume
A7: w is eigenvector of f,L;
    (a*f).w = a*(f.w) by MATRLIN:def 4
      .= a*(L*w) by A7,Def3
      .= (a*L)*w by VECTSP_1:def 16;
    hence w is eigenvector of a*f,a*L by A5,A6,Def3;
  end;
  assume
A8: w is eigenvector of a*f,a*L;
  a*f.w = (a*f).w by MATRLIN:def 4
    .= (a*L)*w by A5,A6,A8,Def3
    .= a*(L*w) by VECTSP_1:def 16;
  then 0.V = a*f.w+-a*(L*w) by VECTSP_1:16
    .= a*f.w+a*(-(L*w)) by VECTSP_1:22
    .= a*(f.w-(L*w)) by VECTSP_1:def 14;
  then f.w-(L*w)=0.V by A1,VECTSP_1:15;
  then -f.w=-(L*w) by VECTSP_1:16;
  then f.w=L*w by RLVECT_1:18;
  hence thesis by Def3;
end;
