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;

theorem
  j in Seg i & a1 = R1.j & a2 = R2.j implies (R1-R2).j = a1 - a2
proof
  assume j in Seg i;
  then j in Seg len (R1-R2) by CARD_1:def 7;
  then j in dom (R1-R2) by FINSEQ_1:def 3;
  hence thesis by Th32;
end;
