
theorem Th31:
  for R being Abelian add-associative right_zeroed
  right_complementable associative distributive well-unital commutative non
  trivial non empty doubleLoopStr, n being Nat ex P being Function
of Polynom-Ring (Polynom-Ring(n,R)),Polynom-Ring(n+1,R) st P is RingIsomorphism
proof
  let R be Abelian add-associative right_zeroed right_complementable
  associative distributive well-unital commutative non trivial
  doubleLoopStr, n being Nat;
  set PN1R = Polynom-Ring(n+1,R);
  set CPRPNR = the carrier of Polynom-Ring (Polynom-Ring(n,R));
  set CPN1R = the carrier of PN1R;
  set P = upm (n,R);
  now
    let p be object;
    assume p in CPN1R;
    then reconsider p9 = p as Element of CPN1R;
    dom P = CPRPNR & P.(mpu(n,R).p9) = p9 by Th30,FUNCT_2:def 1;
    hence p in rng P by FUNCT_1:def 3;
  end;
  then CPN1R c= rng P;
  then rng P = CPN1R by XBOOLE_0:def 10;
  then P is onto;
  then P is RingEpimorphism;
  then P is RingIsomorphism;
  hence thesis;
end;
