
theorem Th25:
  for L being Field
  for p being non-zero Polynomial of L
  for a being Element of L
  for b being non zero Element of L
  for E being Enumeration of Roots(<%a,b%>*'p)
  holds Sum(BRoots(<%a,b%>)(++)E) = -a/b
  proof
    let L be Field;
    let p be non-zero Polynomial of L;
    let a be Element of L;
    let b be non zero Element of L;
    set q = <%a,b%>;
    let E be Enumeration of Roots(q*'p);
    set B = BRoots(q);
    set F = B(++)E;
A1: len F = len E by Def1;
A2: -a/b in {-a/b} by TARSKI:def 1;
A3: Roots(q) = {-a/b} by Th10;
A4: Roots(q*'p) = Roots(q) \/ Roots(p) by UPROOTS:23;
A5: Roots(q) c= Roots(q) \/ Roots(p) by XBOOLE_1:7;
A6: dom E = dom F by A1,FINSEQ_3:29;
    rng E = Roots(q*'p) by RLAFFIN3:def 1;
    then consider j being object such that
A7: j in dom E and
A8: E.j = -a/b by A2,A3,A4,A5,FUNCT_1:def 3;
    reconsider j as Element of NAT by A7;
A9: 1 <= j by A7,FINSEQ_3:25;
A10: j <= len F by A7,A1,FINSEQ_3:25;
A11: E.j = E/.j by A7,PARTFUN1:def 6;
A12: (B*E).j = B.(E.j) by A7,FUNCT_1:13
     .= multiplicity(<%a,b%>,-a/b) by A8,UPROOTS:def 9
     .= 1 by Th11;
    now
      let i be Element of NAT such that
A13:  i in dom F and
A14:  i <> j;
A15:  1 <= i & i <= len F by A13,FINSEQ_3:25;
A16:  E.i = E/.i by A6,A13,PARTFUN1:def 6;
A17:  (B*E).i = B.(E.i) by A6,A13,FUNCT_1:13
      .= multiplicity(q,E/.i) by A16,UPROOTS:def 9
      .= 0 by A16,A8,A6,A7,A13,A14,FUNCT_1:def 4,Th14;
      thus F/.i = F.i by A13,PARTFUN1:def 6
      .= (B*E).i * (E/.i) by A15,Def1
      .= 0.L by A17,BINOM:12;
    end;
    hence Sum F = F/.j by A6,A7,POLYNOM2:3
    .= F.j by A6,A7,PARTFUN1:def 6
    .= (B*E).j * (E/.j) by A9,A10,Def1
    .= -a/b by A8,A11,A12,BINOM:13;
  end;
