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 Th15:
  for V1 be finite-dimensional VectSp of K, b1,b19 be OrdBasis of
  V1 for f be linear-transformation of V1,V1 holds f is with_eigenvalues & L is
  eigenvalue of f iff Det AutEqMt(f+(-L)*id V1,b1,b19) =0.K
proof
  let V1 be finite-dimensional VectSp of K, b1,b19 be OrdBasis of V1;
  let f be linear-transformation of V1,V1;
A1: dim V1=dim V1;
  hereby
    assume f is with_eigenvalues & L is eigenvalue of f;
    then ker (f+(-L)*id V1) is non trivial by Th14;
    hence Det AutEqMt(f+(-L)*id V1,b1,b19) =0.K by A1,MATRLIN2:50;
  end;
  assume Det AutEqMt(f+(-L)*id V1,b1,b19) =0.K;
  then ker (f+(-L)*id V1) is non trivial by A1,MATRLIN2:50;
  hence thesis by Th14;
end;
