reserve S for 1-sorted,
  i for Element of NAT,
  p for FinSequence,
  X for set;

theorem Th34:
  for X being non empty set, s being FinSequence of bspace(X), x
  being Element of X holds (Sum s)@x = Sum (s@x)
proof
  let X be non empty set, s be FinSequence of bspace(X), x be Element of X;
  set V = bspace(X);
  defpred Q[FinSequence of V] means (Sum ($1))@x = Sum (($1)@x);
A1: Q[<*>V]
  proof
    reconsider z = 0.V as Subset of X;
    set e = <*>V;
A2: z@x = 0.Z_2 by Def3;
    Sum e = 0.V & e@x = <*>Z_2 by Th31,RLVECT_1:43;
    hence thesis by A2,RLVECT_1:43;
  end;
A3: for p being FinSequence of V, f being Element of V st Q[p] holds Q[p ^
  <*f*>]
  proof
    let p be FinSequence of V, f be Element of V such that
A4: Q[p];
    (Sum (p ^ <*f*>))@x = ((Sum p) + (Sum <*f*>))@x by RLVECT_1:41
      .= ((Sum p) + f)@x by RLVECT_1:44
      .= (Sum p)@x + f@x by Th32
      .= Sum (p@x) + Sum (<*f@x*>) by A4,RLVECT_1:44
      .= Sum (p@x ^ <*f@x*>) by RLVECT_1:41
      .= Sum ((p ^ <*f*>)@x) by Th33;
    hence thesis;
  end;
  for p being FinSequence of V holds Q[p] from IndSeqS(A1,A3);
  hence thesis;
end;
