
theorem
  for R being add-associative non empty addLoopStr, I,J,K being Subset
  of R holds I + (J + K) = (I + J) + K
proof
  let R be add-associative non empty addLoopStr, I,J,K be Subset of R;
A1: now
    let u be object;
    assume u in (I + J) + K;
    then consider a,b being Element of R such that
A2: u = a + b and
A3: a in I+J and
A4: b in K;
    consider c,d being Element of R such that
A5: a = c + d and
A6: c in I and
A7: d in J by A3;
    d + b in {a9 + b9 where a9,b9 is Element of R : a9 in J & b9 in K} by A4,A7
;
    then
    c+(d + b) in {a9 + b9 where a9,b9 is Element of R : a9 in I & b9 in J
    +K} by A6;
    hence u in I + (J + K) by A2,A5,RLVECT_1:def 3;
  end;
  now
    let u be object;
    assume u in I + (J + K);
    then consider a,b being Element of R such that
A8: u = a + b and
A9: a in I and
A10: b in J+K;
    consider c,d being Element of R such that
A11: b = c + d and
A12: c in J and
A13: d in K by A10;
    a + c in {a9 + b9 where a9,b9 is Element of R : a9 in I & b9 in J} by A9
,A12;
    then (a+ c) + d in {a9 + b9 where a9,b9 is Element of R : a9 in I+J & b9
    in K}by A13;
    hence u in (I + J) + K by A8,A11,RLVECT_1:def 3;
  end;
  hence thesis by A1,TARSKI:2;
end;
