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 Th31:
  for K being Abelian right_zeroed add-associative
right_complementable non empty addLoopStr, R1,R2 being Element of i-tuples_on
  the carrier of K holds -(R1 + R2) = -R1 + -R2
proof
  let K be Abelian right_zeroed add-associative right_complementable non
  empty addLoopStr, R1,R2 be Element of i-tuples_on the carrier of K;
  (R1 + R2) + (-R1 + -R2) = R1 + R2 + -R1 + -R2 by FINSEQOP:28
    .= R2 + R1 + -R1 + -R2 by FINSEQOP:33
    .= R2 + (R1 + -R1) + -R2 by FINSEQOP:28
    .= R2 + (i|->(0.K)) + -R2 by Lm3
    .= R2 + -R2 by Lm2
    .= (i|->0.K) by Lm3;
  hence thesis by Th27;
end;
