theorem
  for K being Abelian add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr,
      a being Element of K,
      p being FinSequence of the carrier of K holds Sum(a*p) = a*(Sum p)
proof
  let K be Abelian add-associative right_zeroed distributive
  right_complementable non empty doubleLoopStr, a be Element of K, p be
  FinSequence of the carrier of K;
  set rM = (the multF of K)[;](a,id the carrier of K);
  the addF of K is having_a_unity & the multF of K is_distributive_wrt the
  addF of K by Th8,Th10;
  hence Sum (a*p) = rM.(Sum p) by Th14,SETWOP_2:30
    .= a*(Sum p) by Lm1;
end;
