
theorem Th14:
  for X being set, L being add-associative right_zeroed
right_complementable non empty addLoopStr, f being Series of X,L holds 0_(X,L
  ) - f = -f
proof
  let X be set, L be add-associative right_zeroed right_complementable non
  empty addLoopStr, f be Series of X,L;
  set p = 0_(X,L) - f;
A1: now
    let x be object;
    assume x in dom p;
    then reconsider b = x as Element of Bags X;
    p.b = (0_(X,L) + -f).b by POLYNOM1:def 7
      .= 0_(X,L).b + (-f).b by POLYNOM1:15
      .= 0.L + (-f).b by POLYNOM1:22
      .= (-f).b by ALGSTR_1:def 2;
    hence p.x = (-f).x;
  end;
  dom p = Bags X by FUNCT_2:def 1
    .= dom(-f) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
