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 K be algebraic-closed Field, V1 be non trivial finite-dimensional
  VectSp of K, f be linear-transformation of V1,V1 holds f is with_eigenvalues
proof
  let K be algebraic-closed Field, V1 be non trivial finite-dimensional VectSp
  of K, f be linear-transformation of V1,V1;
  set b1 = the OrdBasis of V1;
  set AutA=AutMt(f,b1,b1);
  consider P be Polynomial of K such that
A1: len P = len b1+1 and
A2: for x be Element of K holds eval(P,x) = Det(AutA+x*1.(K,len b1)) by Th8;
  dim V1<>0 & dim V1 = len b1 by MATRLIN2:21,42;
  then len P>1+0 by A1,XREAL_1:8;
  then P is with_roots by POLYNOM5:def 9;
  then consider L be Element of K such that
A3: L is_a_root_of P by POLYNOM5:def 8;
A4: Mx2Tran(L*AutMt(id V1,b1,b1),b1,b1) = L*Mx2Tran(AutMt(id V1,b1,b1),b1,b1
  ) by MATRLIN2:38
    .= L*id V1 by MATRLIN2:34
    .= Mx2Tran(AutMt(L*id V1,b1,b1),b1,b1) by MATRLIN2:34;
  0.K = eval(P,L) by A3,POLYNOM5:def 7
    .= Det(AutA+L*1.(K,len b1)) by A2
    .= Det(AutA+L*AutMt(id V1,b1,b1)) by MATRLIN2:28
    .= Det(AutA+AutMt(L*id V1,b1,b1)) by A4,MATRLIN2:39
    .= Det(AutMt(f+L*id V1,b1,b1)) by MATRLIN:42
    .= Det(AutMt(f+(-(-L))*id V1,b1,b1)) by RLVECT_1:17
    .= Det(AutEqMt(f+(-(-L))*id V1,b1,b1)) by MATRLIN2:def 2;
  hence thesis by Th15;
end;
