reserve i,j,k for Nat;
reserve K for non empty addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for left_zeroed right_zeroed add-associative right_complementable
  non empty addLoopStr,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty addLoopStr,
  a1,a2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for Abelian right_zeroed add-associative right_complementable non
  empty addLoopStr,
  R,R1,R2,R3 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a,a9,a1,a2 for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for distributive non empty doubleLoopStr,
  a,a1,a2 for Element of K ,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a1,a2,b1,b2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative non empty multMagma,
  p,q for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative associative non empty multMagma,
  a,a1,a2 for Element of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for commutative associative non empty multMagma,
  a for Element of K,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for add-associative right_zeroed right_complementable non empty
  addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K;

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;
