
theorem Th17:
  for n being set,
      L being add-associative right_zeroed right_complementable
      non empty addLoopStr,
      p being Series of n, L
  for x being bag of n holds (-p).x = -(p.x)
  proof
    let n be set;
    let L be add-associative right_zeroed right_complementable
    non empty addLoopStr;
    let p be Series of n, L;
    let x be bag of n;
A1: dom p = Bags n by FUNCT_2:def 1;
A2: x in Bags n by PRE_POLY:def 12;
    then
A3: p/.x = p.x by A1,PARTFUN1:def 6;
A4: dom (-p) = dom p by VFUNCT_1:def 5;
    hence (-p).x = (-p)/.x by A1,A2,PARTFUN1:def 6
    .= -p.x by A1,A2,A3,A4,VFUNCT_1:def 5;
  end;
