
theorem Th8:
  for L being add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr for x being Element of L for F being
  FinSequence of L holds x * Sum(F) = Sum(x*F)
proof
  let L be add-associative right_zeroed right_complementable distributive non
  empty doubleLoopStr;
  let x be Element of L;
  let F be FinSequence of L;
  defpred P[Nat] means for x being Element of L, F being FinSequence of L st
  len F = $1 holds x * Sum(F) = Sum(x*F);
A1: ex n be Element of NAT st len F = n;
A2: now
    let k be Nat;
    assume
A3: P[k];
    now
      let x be Element of L;
      let F be FinSequence of L;
      set G = F|(Seg k);
      reconsider G as FinSequence by FINSEQ_1:15;
      assume
A4:   len F = k+1;
      then reconsider G as FinSequence of L by Lm1;
A5:   len G = k by A4,Lm1;
A6:   F = G^<*F/.(k+1)*> by A4,Lm1;
      thus x * Sum(F) = x * Sum(G^<*F/.(k+1)*>) by A4,Lm1
        .= x * (Sum G + Sum<*F/.(k+1)*>) by RLVECT_1:41
        .= x * Sum G + x * Sum<*F/.(k+1)*> by VECTSP_1:def 2
        .= Sum(x * G) + x * Sum<*F/.(k+1)*> by A3,A5
        .= Sum(x * G) + x * F/.(k+1) by RLVECT_1:44
        .= Sum(x * G) + Sum(<*x*F/.(k+1)*>) by RLVECT_1:44
        .= Sum(x * G) + Sum(x*<*F/.(k+1)*>) by POLYNOM1:8
        .= Sum((x * G)^(x*<*F/.(k+1)*>)) by RLVECT_1:41
        .= Sum(x*F) by A6,POLYNOM1:10;
    end;
    hence P[k+1];
  end;
  now
    let x be Element of L, F be FinSequence of L;
    assume
A7: len F = 0;
    Seg(len(x*F)) = dom(x*F) by FINSEQ_1:def 3
      .= dom F by POLYNOM1:def 1
      .= Seg len F by FINSEQ_1:def 3;
    then len(x*F) = 0 by A7;
    then
A8: (x*F) = <*>the carrier of L;
    F = <*>the carrier of L by A7;
    then Sum(F) = 0.L by RLVECT_1:43;
    then x * Sum(F) = 0.L;
    hence x * Sum(F) = Sum(x*F) by A8,RLVECT_1:43;
  end;
  then
A9: P[0];
  for k be Nat holds P[k] from NAT_1:sch 2(A9,A2);
  hence thesis by A1;
end;
