
theorem Th13:
  for L being add-associative right_zeroed right_complementable
  right_unital distributive non empty doubleLoopStr, a being Element of L, p
  being FinSequence of the carrier of L holds Sum (p*a) = (Sum p)*a
proof
  let L be add-associative right_zeroed right_complementable right_unital
  distributive non empty doubleLoopStr, a be Element of L;
  set p = <*>(the carrier of L);
  defpred P[FinSequence of the carrier of L] means Sum ($1*a) = (Sum $1)*a;
A1: now
    let p be FinSequence of the carrier of L, r be Element of L such that
A2: P[p];
    Sum ((p^<*r*>)*a) = Sum ((p*a)^(<*r*>*a)) by Th11
      .= Sum (p*a) + Sum (<*r*>*a) by RLVECT_1:41
      .= Sum (p*a) + Sum (<*r*a*>) by Th9
      .= Sum (p*a) + r*a by RLVECT_1:44
      .= (Sum p)*a + (Sum<*r*>)*a by A2,RLVECT_1:44
      .= (Sum p + Sum<*r*>)*a by VECTSP_1:def 7
      .= (Sum (p^<*r*>))*a by RLVECT_1:41;
    hence P[p^<*r*>];
  end;
  Sum p = 0.L & Sum (p*a) = Sum p by Th7,RLVECT_1:43;
  then
A3: P[p];
  thus for p being FinSequence of the carrier of L holds P[p] from FINSEQ_2:
  sch 2(A3,A1);
end;
