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 Th32:
  i in dom (p1-p2) & a1 = p1.i & a2 = p2.i implies (p1-p2).i = a1 - a2
proof
  assume i in dom (p1-p2) & a1 = p1.i & a2 = p2.i;
  then (p1 - p2).i = (diffield(K)).(a1,a2) by FUNCOP_1:22;
  hence thesis by Th11;
end;
