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;

theorem Th27:
  R1 + R2 = (i|->0.K) implies R1 = -R2 & R2 = -R1
proof
A1: the addF of K is having_an_inverseOp & the_inverseOp_wrt the addF of K =
  ( comp K) by Th14,Th15;
  the_unity_wrt the addF of K = 0.K & the addF of K is having_a_unity by Th7
,Th8;
  hence thesis by A1,FINSEQOP:74;
end;
