
theorem Th26:
for L being Abelian add-associative right_zeroed right_complementable
            well-unital distributive non degenerated doubleLoopStr
for p being even Polynomial of L
for q being odd Polynomial of L
for x being Element of L
st x is_a_common_root_of p,q holds -x is_a_root_of p+q
proof
let L be Abelian add-associative right_zeroed right_complementable
         well-unital distributive non degenerated doubleLoopStr;
let p be even Polynomial of L;
let q be odd Polynomial of L;
let x be Element of L;
assume A1: x is_a_common_root_of p,q;
then A2: eval(p,x) = 0.L by POLYNOM5:def 7;
eval(p+q,-x) = eval(p,-x) + eval(q,-x) by POLYNOM4:19
            .= eval(p,x) + eval(q,-x) by Th24
            .= eval(p,x) + - eval(q,x) by Th25
            .= 0.L + - 0.L by A2,A1,POLYNOM5:def 7
            .= 0.L by RLVECT_1:5;
hence -x is_a_root_of p+q by POLYNOM5:def 7;
end;
