
theorem Th13:
for L being Abelian add-associative right_zeroed right_complementable
            unital distributive non empty doubleLoopStr
for p being Polynomial of L
for x being Element of L
st x is_a_root_of p holds x is_a_root_of (-p)
proof
let L be Abelian add-associative right_zeroed right_complementable
         unital distributive non empty doubleLoopStr;
let p be Polynomial of L;
let x be Element of L;
assume A1: x is_a_root_of p;
eval(-p,x) = - eval(p,x) by POLYNOM4:20
          .= - 0.L by A1,POLYNOM5:def 7
          .= 0.L by RLVECT_1:12;
hence x is_a_root_of -p by POLYNOM5:def 7;
end;
