defpred S1[ Nat] means for B being non zero bag of the carrier of R st card () = \$1 holds
ex p being Ppoly of R st
( deg p = card B & ( for a being Element of R holds multiplicity (p,a) = B . a ) );
IA: S1 by lempolybag;
IS: now :: thesis: for k being Nat st 1 <= k & S1[k] holds
S1[k + 1]
let k be Nat; :: thesis: ( 1 <= k & S1[k] implies S1[k + 1] )
assume AS: 1 <= k ; :: thesis: ( S1[k] implies S1[k + 1] )
assume IV: S1[k] ; :: thesis: S1[k + 1]
now :: thesis: for B being non zero bag of the carrier of R st card () = k + 1 holds
ex p being Ppoly of R st
( deg p = card B & ( for a being Element of R holds multiplicity (p,a) = B . a ) )
let B be non zero bag of the carrier of R; :: thesis: ( card () = k + 1 implies ex p being Ppoly of R st
( deg p = card B & ( for a being Element of R holds multiplicity (p,a) = B . a ) ) )

assume X: card () = k + 1 ; :: thesis: ex p being Ppoly of R st
( deg p = card B & ( for a being Element of R holds multiplicity (p,a) = B . a ) )

now :: thesis: ( ( for x being Element of R holds not x in support B ) implies for o being Element of support B holds contradiction )
assume A: for x being Element of R holds not x in support B ; :: thesis: for o being Element of support B holds contradiction
let o be Element of support B; :: thesis: contradiction
hence contradiction by X; :: thesis: verum
end;
then consider x being Element of R such that
A: x in support B ;
H1: for o being object st o in {x} holds
o in support B by ;
set b = ({x},(B . x)) -bag ;
set b1 = B \ x;
B . x <> 0 by ;
then support (({x},(B . x)) -bag) = {x} by UPROOTS:8;
then card (support (({x},(B . x)) -bag)) = 1 by CARD_1:30;
then consider p1 being Ppoly of R such that
A1: ( deg p1 = card (({x},(B . x)) -bag) & ( for a being Element of R holds multiplicity (p1,a) = (({x},(B . x)) -bag) . a ) ) by lempolybag;
A3: card (support (B \ x)) = card (() \ {x}) by bb3a
.= (card ()) - () by
.= (card ()) - 1 by CARD_1:30 ;
then support (B \ x) <> {} by X, AS;
then reconsider b1 = B \ x as non zero bag of the carrier of R by bbag;
consider p2 being Ppoly of R such that
A5: ( deg p2 = card b1 & ( for a being Element of R holds multiplicity (p2,a) = b1 . a ) ) by A3, X, IV;
reconsider q = p1 *' p2 as Ppoly of R by lemppoly3;
( p1 <> 0_. R & p2 <> 0_. R ) ;
then A2: deg q = (deg p1) + (deg p2) by HURWITZ:23
.= card ((({x},(B . x)) -bag) + b1) by
.= card B by bb3 ;
now :: thesis: for a being Element of R holds multiplicity (q,a) = B . a
let a be Element of R; :: thesis: multiplicity (q,a) = B . a
thus multiplicity (q,a) = (multiplicity (p1,a)) + (multiplicity (p2,a)) by UPROOTS:55
.= ((({x},(B . x)) -bag) . a) + (multiplicity (p2,a)) by A1
.= ((({x},(B . x)) -bag) . a) + (b1 . a) by A5
.= ((({x},(B . x)) -bag) + b1) . a by PRE_POLY:def 5
.= B . a by bb3 ; :: thesis: verum
end;
hence ex p being Ppoly of R st
( deg p = card B & ( for a being Element of R holds multiplicity (p,a) = B . a ) ) by A2; :: thesis: verum
end;
hence S1[k + 1] ; :: thesis: verum
end;
I: for k being Nat st 1 <= k holds
S1[k] from NAT_1:sch 8(IA, IS);
consider n being Nat such that
H: card () = n ;
now :: thesis: not n = 0
assume n = 0 ; :: thesis: contradiction
then support B = {} by H;
hence contradiction by bbag; :: thesis: verum
end;
then n + 1 > 0 + 1 by XREAL_1:6;
then 1 <= n by NAT_1:13;
hence ex b1 being Ppoly of R st
( deg b1 = card B & ( for a being Element of R holds multiplicity (b1,a) = B . a ) ) by H, I; :: thesis: verum