reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;

theorem Th8:
  for A be Matrix of n,K ex P be Polynomial of K st len P = n+1 &
  for x be Element of K holds eval(P,x) = Det(A+x*1.(K,n))
proof
  defpred P[Nat] means for A be Matrix of $1,K ex P be Polynomial of K st len
  P = $1+1 & for x be Element of K holds eval(P,x) = Det(A+x*1.(K,$1));
A1: for n st P[n] holds P[n+1]
  proof
    let n such that
A2: P[n];
    set n1=n+1;
    let A be Matrix of n1,K;
    set ONE=1.(K,n1);
A3: Indices ONE=Indices A by MATRIX_0:26;
    defpred Q[Nat] means 1<= $1 & $1 <= n1 implies ex P be Polynomial of K st
len P = n1+1 & for x be Element of K holds eval(P,x) = Sum (LaplaceExpL(A+x*1.(
    K,n1),1)|$1);
A4: n1-'1=n by NAT_D:34;
A5: 1<=n1 by NAT_1:11;
    then
A6: 1 in Seg n1;
A7: Indices A=[:Seg n1,Seg n1:] by MATRIX_0:24;
A8: for k st Q[k] holds Q[k+1]
    proof
      let k such that
A9:   Q[k];
      set k1=k+1;
      assume that
A10:  1<=k1 and
A11:  k1<=n1;
      set pow=power(K).(-1_K,k1+1);
      set P2=<% A*(1,k1)*pow,ONE*(1,k1)*pow %>;
A12:  k1 in Seg n1 by A10,A11;
      then
A13:  [1,k1] in Indices A by A7,A6,ZFMISC_1:87;
      per cases by NAT_1:14;
      suppose
A14:    k=0;
        then pow = (-1_K) * (-1_K) by GROUP_1:51
          .= 1_K * 1_K by VECTSP_1:10
          .= 1_K;
        then
A15:    ONE*(1,k1)*pow = 1_K*1_K by A3,A13,A14,MATRIX_1:def 3
          .= 1_K;
        then
A16:    2-'1=2-1 & P2.1 <>0.K by POLYNOM5:38,XREAL_1:233;
        ONE*(1,k1)*pow<>0.K by A15;
        then
A17:    len P2=2 by POLYNOM5:40;
        consider P be Polynomial of K such that
A18:    len P = n1 and
A19:    for x be Element of K holds eval(P,x) = Det(Delete(A,1,1)+x*
        1.(K,n)) by A2,A4;
        take PP=P2*'P;
        P.n <> 0.K by A18,ALGSEQ_1:10;
        then P2.(len P2-'1)*P.(len P-'1) <>0.K by A4,A18,A17,A16,VECTSP_1:12;
        hence len PP = n1+2-1 by A18,A17,POLYNOM4:10
          .= n1+1;
        let x be Element of K;
A20:    Delete(A,1,1)+x*1.(K,n) = Delete(A,1,1)+x*Delete(ONE,1,1) by A6,A4,Th6
          .= Delete(A,1,1)+Delete(x*ONE,1,1) by A12,A14,Th5
          .= Delete(A+x*ONE,1,1) by A6,Th4;
A21:    A*(1,k1)*pow+ONE*(1,k1)*pow*x = A*(1,k1)*pow+ONE*(1,k1)*x*pow by
GROUP_1:def 3
          .= (A*(1,k1)+ONE*(1,k1)*x)*pow by VECTSP_1:def 2
          .= (A*(1,k1)+(x*ONE)*(1,k1))*pow by A3,A13,MATRIX_3:def 5
          .= (A+x*ONE)*(1,k1)*pow by A13,MATRIX_3:def 3;
        set L=LaplaceExpL(A+x*ONE,1);
A22:    dom L=Seg len L by FINSEQ_1:def 3;
A23:    len L=n1 by LAPLACE:def 7;
A24:    eval(P2*'P,x) = eval(P2,x)*eval(P,x) by POLYNOM4:24
          .= eval(P2,x) * Det(Delete(A,1,1)+x*1.(K,n)) by A19
          .= Minor(A+x*ONE,1,1)*((A+x*ONE)*(1,1)*pow) by A14,A21,A20,
POLYNOM5:44
          .= (A+x*ONE)*(1,1) * Cofactor(A+x*ONE,1,1) by A14,GROUP_1:def 3
          .= L.1 by A6,A22,A23,LAPLACE:def 7;
        1 <= n1 by A10,A11,XXREAL_0:2; then
        1 <= len L by A23; then
        1 in dom L by FINSEQ_3:25; then
        L.1 = L/.1 by PARTFUN1:def 6;
        hence Sum (L|k1) = Sum <*L/.1*> by A14,A23,CARD_1:27,FINSEQ_5:20
          .= Sum <* eval(P2*'P,x) *> by A6,A22,A23,A24,PARTFUN1:def 6
          .= eval(PP,x) by FINSOP_1:11;
      end;
      suppose
A25:    k>=1;
        consider P1 be Polynomial of K such that
A26:    len P1 <= n+1 and
A27:    for x be Element of K holds eval(P1,x)=Det(Delete(A,1,k1)+x*
        Delete(ONE,1,k1))by A4,Th7;
        consider P be Polynomial of K such that
A28:    len P = n1+1 and
A29:    for x be Element of K holds eval(P,x) = Sum (LaplaceExpL(A+x*
        1.(K,n1),1)|k) by A9,A11,A25,NAT_1:13;
        take PP=P+(A*(1,k1)*pow)*P1;
        A*(1,k1)*pow=0.K or A*(1,k1)*pow<>0.K;
        then len ((A*(1,k1)*pow)*P1)<=n1 by A26,POLYNOM5:24,25;
        then
A30:    len ((A*(1,k1)*pow)*P1) < len P by A28,NAT_1:13;
        hence len PP = max(len ((A*(1,k1)*pow)*P1),len P) by POLYNOM4:7
          .= n1+1 by A28,A30,XXREAL_0:def 10;
        let x be Element of K;
        set L=LaplaceExpL(A+x*ONE,1);
A31:    dom L=Seg len L & len L=n1 by FINSEQ_1:def 3,LAPLACE:def 7;
        then
A32:    L|k1 = (L|k)^<*L.k1*> by A12,FINSEQ_5:10;
A33:    k1<>1 by A25,NAT_1:13;
A34:    A*(1,k1)*pow = (A*(1,k1)+0.K)*pow by RLVECT_1:def 4
          .= (A*(1,k1)+x*0.K)*pow
          .= (A*(1,k1)+x*(ONE*(1,k1)))*pow by A3,A13,A33,MATRIX_1:def 3
          .= (A*(1,k1)+(x*ONE)*(1,k1))*pow by A3,A13,MATRIX_3:def 5
          .= (A+x*ONE)*(1,k1)*pow by A13,MATRIX_3:def 3;
A35:    Delete(A,1,k1)+x*Delete(ONE,1,k1) = Delete(A,1,k1)+Delete(x*ONE,1
        ,k1) by A6,A12,Th5
          .= Delete(A+x*ONE,1,k1) by A6,A12,Th4;
        eval((A*(1,k1)*pow)*P1,x) = (A*(1,k1)*pow)*eval(P1,x) by POLYNOM5:30
          .= Minor(A+x*ONE,1,k1)*((A+x*ONE)*(1,k1)*pow) by A27,A34,A35
          .= (A+x*ONE)*(1,k1)*Cofactor(A+x*ONE,1,k1) by GROUP_1:def 3
          .= L.k1 by A12,A31,LAPLACE:def 7;
        hence Sum (L|k1) = Sum (L|k) + eval((A*(1,k1)*pow)*P1,x) by A32,
FVSUM_1:71
          .= eval(P,x)+eval((A*(1,k1)*pow)*P1,x) by A29
          .= eval(PP,x) by POLYNOM4:19;
      end;
    end;
A36: Q[0];
    for k holds Q[k] from NAT_1:sch 2(A36,A8);
    then consider P be Polynomial of K such that
A37: len P = n1+1 and
A38: for x be Element of K holds eval(P,x) = Sum (LaplaceExpL(A+x*1.(K
    ,n1),1)|n1) by A5;
    take P;
    thus len P=n1+1 by A37;
    let x be Element of K;
    set L=LaplaceExpL(A+x*1.(K,n1),1);
    len L=n1 by LAPLACE:def 7;
    hence eval(P,x) = Sum (L|(len L)) by A38
      .= Sum L by FINSEQ_1:58
      .= Det (A+x*1.(K,n1)) by A6,LAPLACE:25;
  end;
A39: P[0]
  proof
    let A be Matrix of 0,K;
    take P=1_.(K);
    thus len P =0+1 by POLYNOM4:4;
    let x be Element of K;
    thus Det(A+x*1.(K,0)) = 1_K by MATRIXR2:41
      .= eval(P,x) by POLYNOM4:18;
  end;
  for n holds P[n] from NAT_1:sch 2(A39,A1);
  hence thesis;
end;
