reserve G for Abelian add-associative right_complementable right_zeroed
  non empty addLoopStr;
reserve GS for non empty addLoopStr;

theorem
  the addF of GS is commutative associative &
  0.GS is_a_unity_wrt the addF of GS &
  comp GS is_an_inverseOp_wrt the addF of GS implies GS is AbGroup
proof
  assume that
A1: the addF of GS is commutative & the addF of GS is associative and
A2: 0.GS is_a_unity_wrt the addF of GS and
A3: comp GS is_an_inverseOp_wrt the addF of GS;
A4: GS is right_complementable
  proof
    let x be Element of GS;
    reconsider b = (comp GS).x as Element of GS;
    take b;
    thus x+b = the_unity_wrt the addF of GS by A3
      .= (0.GS) by A2,BINOP_1:def 8;
  end;
  for x,y,z being Element of GS holds x+y = y+x & (x+y)+z = x+(y+z) & x+(
  0.GS) = x by A1,A2,BINOP_1:3;
  hence thesis by A4,RLVECT_1:def 2,def 3,def 4;
end;
