
theorem
  for L being algebraic-closed Field
  for p,q being non-zero Polynomial of L st
  len p >= 2 holds SumRoots(p*'q) = SumRoots(p) + SumRoots(q)
  proof
    let L be algebraic-closed Field;
    let p,q 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 f being non-zero Polynomial of L st $1 = len f holds
    SumRoots(f*'q) = SumRoots(f) + SumRoots(q);
A2: P[2]
    proof
      let f be non-zero Polynomial of L;
      assume len f = 2;
      then consider a being Element of L, b being non zero Element of L
      such that
A3:   f = <%a,b%> by Th6;
      SumRoots f = -a/b by A3,Th27;
      hence thesis by A3,Th28;
    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 >= 2 by NAT_2:29;
      k+1 >= k by NAT_1:11;
      then k+1 >= 2 by A7,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
A8:   r is_a_root_of p;
      set P = poly_quotient(p,r);
A9:  len P + 1 = len p by A6,A8,UPROOTS:def 7;
      then reconsider P as non-zero Polynomial of L by A6,UPROOTS:17;
A10:  p = <%-r,1.L%>*'P by A8,UPROOTS:50;
      then
A11:  SumRoots(p) = -(-r)/1.L + SumRoots(P) by Th28;
      p*'q = <%-r,1.L%>*'(P*'q) by A10,POLYNOM3:33;
      hence SumRoots(p*'q) = -(-r)/1.L + SumRoots(P*'q) by Th28
      .= r + (SumRoots(P) + SumRoots(q)) by A5,A6,A9
      .= SumRoots(p) + SumRoots(q) by A11,RLVECT_1:def 3;
    end;
    for k being non trivial Nat holds P[k] from NAT_2:sch 2(A2,A4);
    hence thesis by A1;
  end;
