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
  for K being Abelian right_zeroed add-associative right_complementable
  non empty addLoopStr, R,R1,R2 being Element of i-tuples_on the carrier of K
  holds R1 + R = R2 + R or R1 + R = R + R2 implies R1 = R2
proof
  let K be Abelian right_zeroed add-associative right_complementable non
  empty addLoopStr, R,R1,R2 be Element of i-tuples_on the carrier of K;
  R1 + R = R2 + R iff R1 + R = R + R2 by FINSEQOP:33;
  hence thesis by Lm4;
end;
