
theorem Th16:
for L being add-associative right_zeroed right_complementable
            Abelian non empty addLoopStr,
    p,q being Polynomial of L
holds odd_part(p+q) = odd_part(p) + odd_part(q)
proof
let L be add-associative right_zeroed right_complementable
         Abelian non empty addLoopStr;
let p,q be Polynomial of L;
set opq = odd_part(p+q),
    op = odd_part(p),
    oq = odd_part(q);
A1: dom opq = NAT by FUNCT_2:def 1 .= dom(op + oq) by FUNCT_2:def 1;
now let x be object;
  assume x in dom(opq);
  then reconsider i = x as Element of NAT by FUNCT_2:def 1;
  now per cases;
  case A2: i is odd;
     thus(op + oq).i = op.i + oq.i by NORMSP_1:def 2
                    .= p.i + oq.i by A2,Def2
                    .= p.i + q.i by A2,Def2
                    .= (p+q).i by NORMSP_1:def 2
                    .= (opq).i by A2,Def2;
    end;
  case A3: i is even;
     thus (op + oq).i = op.i + oq.i by NORMSP_1:def 2
                     .= 0.L + oq.i by A3,Def2
                     .= 0.L + 0.L by A3,Def2
                     .= 0.L by RLVECT_1:def 4
                     .= (opq).i by A3,Def2;
    end;
  end;
  hence (op + oq).x = (opq).x;
  end;
hence thesis by A1,FUNCT_1:2;
end;
