reserve F for Field,
  x for Element of F,
  V for VectSp of F,
  v for Element of V;

theorem
  for B being AbGroup holds multMagma (# the carrier of B, the addF of B
  #) is commutative Group
proof
  let B be AbGroup;
  set G = multMagma (# the carrier of B, the addF of B #);
A1: for a,b be Element of G, x,y be Element of B st a = x & b = y holds a *
  b = x + y;
A2: G is associative Group-like
  proof
    reconsider e = 0.B as Element of G;
    thus for a,b,c being Element of G holds a * b * c = a * (b * c)
    proof
      let a,b,c be Element of G;
      reconsider x = a, y = b, z = c as Element of B;
      thus a * b * c = x + y + z .= x + (y + z) by RLVECT_1:def 3
        .= a * (b * c);
    end;
    take e;
    let a be Element of G;
    reconsider x = a as Element of B;
    thus a * e = x + 0.B .= a by RLVECT_1:4;
    reconsider b = - x as Element of G;
    thus e * a = x + 0.B by A1
      .= a by RLVECT_1:4;
    take b;
    thus a * b = x + (- x) .= e by RLVECT_1:5;
    thus b * a = x + (- x) by A1
      .= e by RLVECT_1:5;
  end;
  now
    let a,b be Element of G;
    reconsider x = a, y = b as Element of B;
    thus a * b = y + x by A1
      .= b * a;
  end;
  hence thesis by A2,GROUP_1:def 12;
end;
