reserve i,j,k for Nat;

theorem
  for K being non empty addLoopStr, a1,a2 being Element of K, R1,R2
being Element of i-tuples_on the carrier of K holds j in Seg i & a1 = R1.j & a2
  = R2.j implies (R1+R2).j = a1 + a2
proof
  let K be non empty addLoopStr, a1,a2 be Element of K, R1,R2 be Element of i
  -tuples_on the carrier of K;
  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 FUNCOP_1:22;
end;
