
theorem Th34:
  for L being add-associative right_zeroed right_complementable
  left-distributive non empty doubleLoopStr, a, b being Element of L, p being
Polynomial of L holds (<%a, b%>*'p).0 = a*p.0 & for i being Nat holds (<%a, b%>
  *'p).(i+1) = a*p.(i+1)+b*p.i
proof
  let L be add-associative right_zeroed right_complementable left-distributive
  non empty doubleLoopStr, a, b be Element of L, q be Polynomial of L;
  set p = <%a, b%>;
  consider r being FinSequence of L such that
A1: len r = 0 qua Nat+1 and
A2: p*'q.0 = Sum r and
A3: for k be Element of NAT st k in dom r holds r.k = p.(k-'1) * q.(0
  qua Nat+1-'k) by POLYNOM3:def 9;
A4: 1 in dom r by A1,FINSEQ_3:25;
  then reconsider r1 = r.1 as Element of L by FINSEQ_2:11;
  r = <*r1*> by A1,FINSEQ_1:40;
  then Sum r = r1 by RLVECT_1:44
    .= p.(1-'1) * q.(0 qua Nat+1-'1) by A3,A4
    .= p.0 * q.(0 qua Nat+1-'1) by XREAL_1:232
    .= p.0 * q.0 by NAT_D:34;
  hence (<%a, b%>*'q).0 = a*q.0 by A2,POLYNOM5:38;
  let i be Nat;
  consider r being FinSequence of L such that
A5: len r = (i+1)+1 and
A6: p*'q.(i+1) = Sum r and
A7: for k be Element of NAT st k in dom r holds r.k = p.(k-'1) * q.((i+1
  )+1-'k) by POLYNOM3:def 9;
  len r = i+2 by A5;
  then
A8: 0 qua Nat+2 <= len r by XREAL_1:6;
  then
A9: 2 in dom r by FINSEQ_3:25;
  then
A10: r/.2 = r.2 by PARTFUN1:def 6
    .= p.(1+1-'1) * q.((i+1)+1-'2) by A7,A9
    .= p.1 * q.((i+1)+1-'2) by NAT_D:34
    .= b * q.(i+(1+1)-'2) by POLYNOM5:38
    .= b * q.i by NAT_D:34;
A11: now
    let k be Element of NAT such that
A12: 2 < k and
A13: k in dom r;
    consider k1 being Nat such that
A14: k = k1+1 by A12,NAT_1:6;
A15: 2 <= k1 by A12,A14,NAT_1:13;
    reconsider k1 as Element of NAT by ORDINAL1:def 12;
    thus r.k = p.(k-'1) * q.((i+1)+1-'k) by A7,A13
      .= p.k1 * q.((i+1)+1-'k) by A14,NAT_D:34
      .= (0.L) * q.((i+1)+1-'k) by A15,POLYNOM5:38
      .= 0.L;
  end;
  1 <= len r by A8,XXREAL_0:2;
  then
A16: 1 in dom r by FINSEQ_3:25;
  then r/.1 = r.1 by PARTFUN1:def 6
    .= p.(1-'1) * q.((i+1)+1-'1) by A7,A16
    .= p.0 * q.((i+1)+1-'1) by XREAL_1:232
    .= p.0 * q.(i+1) by NAT_D:34
    .= a*q.(i+1) by POLYNOM5:38;
  hence thesis by A6,A8,A10,A11,Th2;
end;
