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 L be eigenvalue of id V holds L = 1_K
proof
  let L be eigenvalue of id V;
  id V is with_eigenvalues by Th11;
  then consider v be Vector of V such that
A1: v<>0.V and
A2: id V.v = L*v by Def2;
  L*v = v by A2
    .= 1_K*v;
  hence thesis by A1,VECTSP10:4;
end;
