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;

theorem
  for K being add-associative right_zeroed right_complementable
     commutative left_unital distributive non empty doubleLoopStr,
      R being Element of i-tuples_on the carrier of K holds
    (-1.K) * R = -R
proof
  let K be add-associative right_zeroed right_complementable commutative
     left_unital distributive non empty doubleLoopStr,
      R be Element of i-tuples_on the carrier of K;
A1: (comp K).(1.K) = -1.K & the_unity_wrt the multF of K = 1.K
     by Th5,VECTSP_1:def 13;
A2: the addF of K is having_an_inverseOp & the_inverseOp_wrt the addF of K =
  (comp K) by Th14,Th15;
  the multF of K is having_a_unity & the addF of K is having_a_unity by Th8;
  hence thesis by A1,A2,Th10,FINSEQOP:68;
end;
