
theorem
for L being Abelian add-associative right_zeroed right_complementable
            unital distributive non empty doubleLoopStr
for p,q being 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
         unital distributive non empty doubleLoopStr;
let p,q be Polynomial of L;
let x be Element of L;
assume x is_a_common_root_of p,q;
then A1: x is_a_root_of p & x is_a_root_of q;
then A2: eval(p,x) = 0.L by POLYNOM5:def 7;
A3: eval(q,x) = 0.L by A1,POLYNOM5:def 7;
eval(p+q,x) = 0.L + 0.L by A2,A3,POLYNOM4:19
           .= 0.L by RLVECT_1:def 4;
hence thesis by POLYNOM5:def 7;
end;
