
theorem Th9:
for L being left_zeroed right_zeroed non empty addLoopStr,
    p being Polynomial of L
holds even_part p + odd_part p = p
proof
let L be left_zeroed right_zeroed non empty addLoopStr,
    p be Polynomial of L;
set e = even_part p, o = odd_part p;
A1: dom p = NAT by FUNCT_2:def 1 .= dom(e+o) by FUNCT_2:def 1;
now let x be object;
  assume x in dom p;
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  now per cases;
  case A2: i is even;
    hence e.i + o.i = e.i + 0.L by Def2
                   .= e.i by RLVECT_1:def 4 .= p.i by Def1,A2;
    end;
  case A3: i is odd;
    hence e.i + o.i = 0.L + o.i by Def1
                   .= o.i by ALGSTR_1:def 2 .= p.i by Def2,A3;
    end;
  end;
  hence p.x = (e+o).x by NORMSP_1:def 2;
  end;
hence thesis by A1,FUNCT_1:2;
end;
