
theorem
  for n being set,
      L being add-associative right_zeroed right_complementable
      non empty addLoopStr,
      p,r being Series of n, L
  st for x being bag of n holds r.x = -(p.x) holds
  r = -p
  proof
    let n be set;
    let L be add-associative right_zeroed right_complementable
    non empty addLoopStr;
    let p,r be Series of n, L;
    assume
A1: for x being bag of n holds r.x = -p.x;
    let x be Element of Bags n;
A2: dom (-p) = Bags n by FUNCT_2:def 1;
A3: (-p)/.x = (-p).x;
A4: p/.x = p.x;
    thus r.x = -p.x by A1
    .= (-p).x by A2,A3,A4,VFUNCT_1:def 5;
  end;
