
theorem Th7:
for L being add-associative right_zeroed right_complementable distributive
            Abelian domRing-like non trivial doubleLoopStr
for p1,p2 being Polynomial of L
for p3 being non zero Polynomial of L
st p1 *' p3 = p2 *' p3 holds p1 = p2
proof
let L be add-associative right_zeroed right_complementable distributive
         Abelian domRing-like non trivial doubleLoopStr;
let p1,p2 be Polynomial of L;
let p3 be non zero Polynomial of L;
assume A1: p1 *' p3 = p2 *' p3;
reconsider x1 = p1 as Element of Polynom-Ring(L) by POLYNOM3:def 10;
reconsider x2 = p2 as Element of Polynom-Ring(L) by POLYNOM3:def 10;
reconsider x3 = p3 as Element of Polynom-Ring(L) by POLYNOM3:def 10;
p3 <> 0_.(L); then
A2: x3 <> 0.Polynom-Ring(L) by POLYNOM3:def 10;
A3: x1 * x3 = p2 *' p3 by A1,POLYNOM3:def 10
          .= x2 * x3 by POLYNOM3:def 10;
x3 is right_mult-cancelable by A2,ALGSTR_0:def 37;
hence thesis by A3;
end;
