
theorem Th13:
  for X being set, L being add-associative right_zeroed
  right_complementable non empty addLoopStr holds -(0_(X,L)) = 0_(X,L)
proof
  let X be set, L be add-associative right_zeroed right_complementable non
  empty addLoopStr;
  set o = -(0_(X,L)), e = 0_(X,L);
A1: now
    let x be object;
    assume x in dom o;
    then reconsider b = x as bag of X;
    o.b = -(e.b) by POLYNOM1:17
      .= -(0.L) by POLYNOM1:22
      .= 0.L by RLVECT_1:12
      .= e.b by POLYNOM1:22;
    hence o.x = e.x;
  end;
  dom o = Bags X by FUNCT_2:def 1
    .= dom e by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
