
theorem Th12:
  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 (a*p) = a*Sum p
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 (a*$1) = a*Sum $1;
A1: now
    let p be FinSequence of the carrier of L, r be Element of L such that
A2: P[p];
    Sum (a*(p^<*r*>)) = Sum ((a*p)^(a*<*r*>)) by Th10
      .= Sum (a*p) + Sum (a*<*r*>) by RLVECT_1:41
      .= Sum (a*p) + Sum (<*a*r*>) by Th8
      .= Sum (a*p) + a*r by RLVECT_1:44
      .= a*Sum p + a*Sum<*r*> by A2,RLVECT_1:44
      .= a*(Sum p + Sum<*r*>) by VECTSP_1:def 7
      .= a*Sum (p^<*r*>) by RLVECT_1:41;
    hence P[p^<*r*>];
  end;
  Sum p = 0.L & Sum (a*p) = Sum p by Th6,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;
