
theorem Th13:
  for n being Ordinal, L being add-associative right_zeroed
right_complementable non empty addLoopStr, p being Series of n,L, b being bag
  of n holds b *' (-p) = -(b *' p)
proof
  let n be Ordinal, L be add-associative right_zeroed right_complementable
  non empty addLoopStr, p be Series of n,L, b be bag of n;
  set q1 = b *' (-p), q2 = -(b *' p);
A1: now
    let x be object;
    assume x in dom q1;
    then reconsider u = x as bag of n;
    now
      per cases;
      case
A2:     b divides u;
        then
A3:     (b*'p).u = p.(u-'b) by POLYRED:def 1;
        thus q1.u = (-p).(u-'b) by A2,POLYRED:def 1
          .= -(p.(u-'b)) by POLYNOM1:17
          .= (-(b*'p)).u by A3,POLYNOM1:17;
      end;
      case
A4:     not b divides u;
        then
A5:     (b*'p).u = 0.L by POLYRED:def 1;
        thus q1.u = 0.L by A4,POLYRED:def 1
          .= -0.L by RLVECT_1:12
          .= q2.u by A5,POLYNOM1:17;
      end;
    end;
    hence q1.x = q2.x;
  end;
  dom q1 = Bags n by FUNCT_2:def 1
    .= dom q2 by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
