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 Th11:
  id V is with_eigenvalues & 1_K is eigenvalue of id V & for v
  holds v is eigenvector of id V,1_K
proof
  thus
A1: id V is with_eigenvalues by Lm2;
  ex v st v<>0.V & id V.v=1_K*v by Lm2;
  hence
A2: 1_K is eigenvalue of id V by A1,Def2;
  let w;
  id V.w = w
    .= 1_K*w;
  hence thesis by A1,A2,Def3;
end;
