
theorem
for L being add-associative right_zeroed right_complementable
            Abelian non empty addLoopStr,
    p being Polynomial of L
holds odd_part p - p = - even_part p
proof
let L be add-associative right_zeroed right_complementable
         Abelian non empty addLoopStr,
    p be Polynomial of L;
set e = even_part p, o = odd_part p;
A1: dom(o-p) = NAT by FUNCT_2:def 1 .= dom(-e) by FUNCT_2:def 1;
now let x be object;
  assume x in dom(-e);
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  p = o + e by Th9;
  then p.i = o.i + e.i by NORMSP_1:def 2;
  then o.i - p.i = o.i + (-e.i + - o.i) by RLVECT_1:31
                .= (o.i + -o.i) + - e.i by RLVECT_1:def 3
                .= 0.L + - e.i by RLVECT_1:5
                .= - e.i by RLVECT_1:def 4
                .= (-e).i by BHSP_1:44;
  hence (o-p).x = (-e).x by POLYNOM3:27;
  end;
hence thesis by A1,FUNCT_1:2;
end;
