
theorem
  for L being algebraic-closed Field, p being non-zero Polynomial of L holds
  len p >= 2 implies SumRoots p = - p.(len p-'2) / p.(len p-'1)
  proof
    let L be algebraic-closed Field, p be non-zero Polynomial of L;
    assume len p >= 2;
    then len p <> 0 & len p <> 1;
    then
A1: len p is non trivial by NAT_2:28;
    defpred P[Nat] means
    for p being non-zero Polynomial of L holds
    $1 = len p implies SumRoots p = - p.($1-'2) / p.($1-'1);
A2: P[2]
    proof
      let p be non-zero Polynomial of L;
      assume len p = 2;
      then consider a being Element of L, b being non zero Element of L
      such that
A3:   p = <%a,b%> by Th6;
      p.0 = a & p.1 = b by A3,POLYNOM5:38;
      hence thesis by A3,Lm1,Lm2,Th27;
    end;
A4: for k being non trivial Nat st P[k] holds P[k+1]
    proof
      let k be non trivial Nat such that
A5:   P[k];
      let p being non-zero Polynomial of L such that
A6:   k+1 = len p;
A7:   k+1-'1 = k by NAT_D:34;
      k-1 >= 2-1 by XREAL_1:9,NAT_2:29;
      then
A8:   k+1-'2 = k+1-2 by XREAL_0:def 2
      .= k-1;
      then reconsider k1 = k-1 as Nat;
A9:   k >= 2 by NAT_2:29;
      k+1 >= k by NAT_1:11;
      then k+1 >= 2 by A9,XXREAL_0:2;
      then len p > 1 by A6,XXREAL_0:2;
      then p is with_roots by POLYNOM5:def 9;
      then consider r being Element of L such that
A10:  r is_a_root_of p;
      set P = poly_quotient(p,r);
A11:  len P + 1 = len p by A6,A10,UPROOTS:def 7;
      then reconsider P as non-zero Polynomial of L by A6,UPROOTS:17;
      reconsider k2 = k-2 as Element of NAT by NAT_2:29,INT_1:5;
A12:  k-'2 = k2 by XREAL_0:def 2;
A13:  k-'1 = k1 by XREAL_0:def 2;
A14:  P.k = 0.L by A6,A11,ALGSEQ_1:8;
A15:  p = <%-r,1.L%>*'P by A10,UPROOTS:50;
      then
A16:  p.(k1+1) = (-r)*P.(k1+1)+1.L*P.k1 by UPROOTS:37;
A17:  p.(k2+1) = (-r)*P.(k2+1)+1.L*P.k2 by A15,UPROOTS:37;
      -((-r)*P.k1) = (--r)*P.k1 by VECTSP_1:9;
      then
A18:  -((-r)*P.k1+P.k2) = r*P.k1 - P.k2 by RLVECT_1:30;
A19:  len P = k1+1 by A6,A11;
      then
A20:  P.k1 <> 0.L by ALGSEQ_1:10;
A21:  P.k1*(/P.k1) = (/P.k1)*P.k1;
      P.k1 is non zero by A19,ALGSEQ_1:10;
      then
A22:  r = r*P.k1/P.k1 by A21,Th4;
      thus SumRoots(p) = -(-r)/1.L + SumRoots(P) by A15,Th28
      .= r - P.k2/P.k1 by A5,A6,A11,A12,A13
      .= (r*P.k1-P.k2) / P.k1 by A20,A22,VECTSP_2:20
      .= - p.(k+1-'2) / p.(k+1-'1) by A7,A8,A14,A16,A17,A18,A20,VECTSP_2:19;
    end;
    for k being non trivial Nat holds P[k] from NAT_2:sch 2(A2,A4);
    hence thesis by A1;
  end;
