
theorem Th9:
  for X being set, L being add-associative right_zeroed
right_complementable distributive non empty doubleLoopStr, p being Series of
  X,L, a being Element of L holds -(a * p) = (-a) * p & -(a * p) = a * (-p)
proof
  let n be set, L be add-associative right_zeroed right_complementable
distributive non empty doubleLoopStr, p be Series of n,L, a be Element of L;
  set ap = a * p;
A1: now
    let u be object;
    assume u in dom -ap;
    then reconsider u9 = u as bag of n;
    (-ap).u9 = -(ap.u9) by POLYNOM1:17
      .= -(a * p.u9) by POLYNOM7:def 9
      .= (-a) * p.u9 by VECTSP_1:9
      .= ((-a)*p).u9 by POLYNOM7:def 9;
    hence (-ap).u = ((-a)*p).u;
  end;
  dom -ap = Bags n by FUNCT_2:def 1
    .= dom((-a) * p) by FUNCT_2:def 1;
  hence -(a * p) = (-a) * p by A1,FUNCT_1:2;
A2: now
    let u be object;
    assume u in dom -ap;
    then reconsider u9 = u as bag of n;
    (-ap).u9 = -(ap.u9) by POLYNOM1:17
      .= -(a * p.u9) by POLYNOM7:def 9
      .= a * (-p.u9) by VECTSP_1:8
      .= a * (-p).u9 by POLYNOM1:17
      .= (a*(-p)).u9 by POLYNOM7:def 9;
    hence (-ap).u = (a*(-p)).u;
  end;
  dom -ap = Bags n by FUNCT_2:def 1
    .= dom(a * (-p)) by FUNCT_2:def 1;
  hence thesis by A2,FUNCT_1:2;
end;
