
theorem
for L being add-associative right_zeroed right_complementable
      non empty addLoopStr,
    p being Polynomial of L
holds p - odd_part p = even_part p
proof
let L be add-associative right_zeroed right_complementable
     non empty addLoopStr,
    p be Polynomial of L;
set e = even_part p, o = odd_part p;
A1: dom(p-o) = 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 = e + o by Th9;
  then p.i = e.i + o.i by NORMSP_1:def 2;
  then p.i - o.i = e.i + (o.i + - o.i) by RLVECT_1:def 3
                .= e.i + 0.L by RLVECT_1:5
                .= e.i by RLVECT_1:def 4;
  hence (p-o).x = e.x by POLYNOM3:27;
  end;
hence thesis by A1,FUNCT_1:2;
end;
