set PNR = Polynom-Ring (n,R);
set PN1R = Polynom-Ring ((n + 1),R);
set PRPNR = Polynom-Ring (Polynom-Ring (n,R));
set CPN1R = the carrier of (Polynom-Ring ((n + 1),R));
let A, B be Function of (Polynom-Ring ((n + 1),R)),(Polynom-Ring (Polynom-Ring (n,R))); :: thesis: ( ( for p1 being Polynomial of (n + 1),R
for p2 being Polynomial of n,R
for p3 being Polynomial of (Polynom-Ring (n,R))
for i being Element of NAT
for b being bag of n st p3 = A . p1 & p2 = p3 . i holds
p2 . b = p1 . () ) & ( for p1 being Polynomial of (n + 1),R
for p2 being Polynomial of n,R
for p3 being Polynomial of (Polynom-Ring (n,R))
for i being Element of NAT
for b being bag of n st p3 = B . p1 & p2 = p3 . i holds
p2 . b = p1 . () ) implies A = B )

assume A25: for p1 being Polynomial of (n + 1),R
for p2 being Polynomial of n,R
for p3 being Polynomial of (Polynom-Ring (n,R))
for i being Element of NAT
for b being bag of n st p3 = A . p1 & p2 = p3 . i holds
p2 . b = p1 . () ; :: thesis: ( ex p1 being Polynomial of (n + 1),R ex p2 being Polynomial of n,R ex p3 being Polynomial of (Polynom-Ring (n,R)) ex i being Element of NAT ex b being bag of n st
( p3 = B . p1 & p2 = p3 . i & not p2 . b = p1 . () ) or A = B )

assume A26: for p1 being Polynomial of (n + 1),R
for p2 being Polynomial of n,R
for p3 being Polynomial of (Polynom-Ring (n,R))
for i being Element of NAT
for b being bag of n st p3 = B . p1 & p2 = p3 . i holds
p2 . b = p1 . () ; :: thesis: A = B
now :: thesis: for x being object st x in the carrier of (Polynom-Ring ((n + 1),R)) holds
A . x = B . x
let x be object ; :: thesis: ( x in the carrier of (Polynom-Ring ((n + 1),R)) implies A . x = B . x )
assume A27: x in the carrier of (Polynom-Ring ((n + 1),R)) ; :: thesis: A . x = B . x
then reconsider x99 = x as Element of the carrier of (Polynom-Ring ((n + 1),R)) ;
reconsider x9 = x as Polynomial of (n + 1),R by ;
reconsider Ax = A . x99, Bx = B . x99 as Polynomial of (Polynom-Ring (n,R)) by POLYNOM3:def 10;
now :: thesis: for i being object st i in NAT holds
Ax . i = Bx . i
let i be object ; :: thesis: ( i in NAT implies Ax . i = Bx . i )
assume i in NAT ; :: thesis: Ax . i = Bx . i
then reconsider i9 = i as Element of NAT ;
reconsider Axi = Ax . i9, Bxi = Bx . i9 as Polynomial of n,R by POLYNOM1:def 11;
now :: thesis: for b being object st b in Bags n holds
Axi . b = Bxi . b
let b be object ; :: thesis: ( b in Bags n implies Axi . b = Bxi . b )
assume b in Bags n ; :: thesis: Axi . b = Bxi . b
then reconsider b9 = b as bag of n ;
thus Axi . b = x9 . (b9 bag_extend i9) by A25
.= Bxi . b by A26 ; :: thesis: verum
end;
hence Ax . i = Bx . i by FUNCT_2:12; :: thesis: verum
end;
hence A . x = B . x by FUNCT_2:12; :: thesis: verum
end;
hence A = B ; :: thesis: verum