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 f1,f2 be with_eigenvalues Function of V,V for L1,L2 be Scalar of K
  st L1 is eigenvalue of f1 & L2 is eigenvalue of f2 & ex v st v is eigenvector
of f1,L1 & v is eigenvector of f2,L2 & v<>0.V holds f1+f2 is with_eigenvalues &
  L1+L2 is eigenvalue of f1+f2 & for w st w is eigenvector of f1,L1 & w is
  eigenvector of f2,L2 holds w is eigenvector of f1+f2,L1+L2
proof
  let f1,f2 be with_eigenvalues Function of V,V;
  let L1,L2 be Scalar of K;
  set g=f1+f2;
  assume that
A1: L1 is eigenvalue of f1 and
A2: L2 is eigenvalue of f2;
  given v such that
A3: v is eigenvector of f1,L1 and
A4: v is eigenvector of f2,L2 and
A5: v<>0.V;
A6: g.v = f1.v+f2.v by MATRLIN:def 3
    .= L1*v+f2.v by A1,A3,Def3
    .= L1*v+L2*v by A2,A4,Def3
    .= (L1+L2)*v by VECTSP_1:def 15;
  hence
A7: g is with_eigenvalues by A5;
  hence
A8: L1+L2 is eigenvalue of g by A5,A6,Def2;
  let w such that
A9: w is eigenvector of f1,L1 and
A10: w is eigenvector of f2,L2;
  g.w = f1.w+f2.w by MATRLIN:def 3
    .= L1*w+f2.w by A1,A9,Def3
    .= L1*w+L2*w by A2,A10,Def3
    .= (L1+L2)*w by VECTSP_1:def 15;
  hence thesis by A7,A8,Def3;
end;
