
theorem lemma2z:
for F being Field
for G1,G2 being non empty FinSequence of Polynom-Ring F
st (for i being Nat st i in dom G1
    ex a being Element of F st G1.i = rpoly(1,a)) &
   (for i being Nat st i in dom G2
    ex a being Element of F st G2.i = rpoly(1,a)) &
   Product G1 = Product G2
for a being Element of F holds
(ex i being Nat st i in dom G1 & G1.i = rpoly(1,a)) iff
(ex i being Nat st i in dom G2 & G2.i = rpoly(1,a))
proof
let F be Field;
let G1,G2 be non empty FinSequence of Polynom-Ring F;
assume AS: (for i being Nat st i in dom G1
            ex a being Element of F st G1.i = rpoly(1,a)) &
           (for i being Nat st i in dom G2
            ex a being Element of F st G2.i = rpoly(1,a)) &
           Product G1 = Product G2;
let a be Element of F;
Product G1 is Polynomial of F by POLYNOM3:def 10; then
reconsider p1 = Product G1 as Ppoly of F by AS,RING_5:def 4;
Product G1 <> 0_.(F) by AS,RATFUNC1:11; then
A: Product G1 is non zero by POLYNOM3:def 10;
deg p1 <> 0 by RATFUNC1:def 2; then
B: Product G1 is NonUnit of Polynom-Ring F by RING_4:37;
G1 is Factorization of (Product G1) &
G2 is Factorization of (Product G1) by AS,lemma2y; then
consider B being Function of dom G1, dom G2 such that
C: B is bijective &
   for i being Element of dom G1 holds G2.(B.i) is_associated_to G1.i
   by A,B,RING_2:def 14;
X: now assume
   ex i being Nat st i in dom G1 & G1.i = rpoly(1,a); then
   consider i being Nat such that
   X1: i in dom G1 & G1.i = rpoly(1,a);
   reconsider i as Element of dom G1 by X1;
   consider j being Element of dom G1 such that
   X2: G2.(B.i) is_associated_to G1.i by C;
   consider b being Element of F such that
   X3: G2.(B.i) = rpoly(1,b) by AS;
   reconsider p1 = G1.i, p2 = G2.(B.i) as
                                Element of the carrier of Polynom-Ring F;
   thus ex i being Nat st i in dom G2 & G2.i = rpoly(1,a)
                                                   by X1,X2,X3,RING_4:30;
   end;
now assume
   ex i being Nat st i in dom G2 & G2.i = rpoly(1,a); then
   consider i being Nat such that
   X1: i in dom G2 & G2.i = rpoly(1,a);
   reconsider i as Element of dom G2 by X1;
   X2: B is one-to-one onto by C;
   reconsider C = B" as Function of dom G2, dom G1 by X2,FUNCT_2:25;
   C.i in dom G1; then
   reconsider j = (B").i as Element of dom G1;
   X3: G2.(B.j) is_associated_to G1.j by C;
   consider b being Element of F such that
   X4: G1.j = rpoly(1,b) by AS;
   reconsider p1 = G1.j, p2 = G2.i as
                                Element of the carrier of Polynom-Ring F;
   i = B.j by X2,FUNCT_1:35;
   hence ex i being Nat st i in dom G1 & G1.i = rpoly(1,a)
                                                   by X1,X4,X3,RING_4:30;
   end;
hence thesis by X;
end;
