
theorem Th14:
for L being Abelian add-associative right_zeroed right_complementable
            unital left-distributive non empty doubleLoopStr
for p1,p2 being Polynomial of L
for x being Element of L
st x is_a_common_root_of p1,p2 holds x is_a_root_of p1 + p2
proof
let L be Abelian add-associative right_zeroed right_complementable
         unital left-distributive non empty doubleLoopStr;
let p1,p2 be Polynomial of L;
let x be Element of L;
assume x is_a_common_root_of p1,p2;
then x is_a_root_of p1 & x is_a_root_of p2;
then A1: eval(p1,x) = 0.L & eval(p2,x) = 0.L by POLYNOM5:def 7;
eval(p1+p2,x) = 0.L + 0.L by A1,POLYNOM4:19
             .= 0.L by RLVECT_1:def 4;
hence x is_a_root_of p1+p2 by POLYNOM5:def 7;
end;
