reserve x,y for set;

theorem
  for F being Field holds the addLoopStr of F is AbGroup
proof
  let F be Field;
  set L = the addLoopStr of F;
A1: L is add-associative
  proof
    let u,v,w being Element of L;
    reconsider a=u, b=v, c = w as Element of F;
    thus (u + v) + w = (a + b) + c .= a + (b + c) by RLVECT_1:def 3
      .= u + (v + w);
  end;
A2: L is right_zeroed
  proof
    let v be Element of L;
    reconsider a=v as Element of F;
    thus v + 0.L = a + 0.F .= v;
  end;
A3: L is right_complementable
  proof
    let v be Element of L;
    reconsider a=v as Element of F;
    consider b being Element of F such that
A4: a + b = 0.F by ALGSTR_0:def 11;
    reconsider w=b as Element of L;
    take w;
    thus thesis by A4;
  end;
  L is Abelian
  proof
    let v,w be Element of L;
    reconsider a=v, b=w as Element of F;
    thus v + w = a + b .= b + a
      .= w + v;
  end;
  hence thesis by A1,A2,A3;
end;
