reserve i,j,k for Nat;

theorem Th2:
  for K being add-associative non empty addLoopStr holds
    the addF of K is associative
proof
  let K be add-associative non empty addLoopStr;
  now
    let a,b,c be Element of K;
    thus (the addF of K).(a,(the addF of K).(b,c))=a+(b+c)
      .=(a+b)+c by RLVECT_1:def 3
      .=(the addF of K).((the addF of K).(a,b),c);
  end;
  hence thesis by BINOP_1:def 3;
end;
