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;
reserve K for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  p for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;

theorem
  Sum(R1 - R2) = Sum R1 - Sum R2
proof
  the addF of K is having_an_inverseOp & the_inverseOp_wrt the addF of K =
  (comp K) by Th14,Th15;
  hence Sum(R1 - R2) = (diffield(K)).(Sum R1,(the addF of K)$$R2)
    by Th8,SETWOP_2:37
    .= Sum R1 - Sum R2 by Th11;
end;
