set d = <%1,1,3,1%>;
set e = <%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2)),(1 * (10 |^ 3))%>;
A1: Sum <%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2)),(1 * (10 |^ 3))%> = (Sum <%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2))%>) + (Sum <%(1 * (10 |^ 3))%>) by AFINSQ_2:55
.= ((Sum <%(1 * (10 |^ 0)),(1 * (10 |^ 1))%>) + (Sum <%(3 * (10 |^ 2))%>)) + (Sum <%(1 * (10 |^ 3))%>) by AFINSQ_2:55
.= (((1 * (10 |^ 0)) + (1 * (10 |^ 1))) + (Sum <%(3 * (10 |^ 2))%>)) + (Sum <%(1 * (10 |^ 3))%>) by AFINSQ_2:54
.= (((1 * (10 |^ 0)) + (1 * (10 |^ 1))) + (3 * (10 |^ 2))) + (Sum <%(1 * (10 |^ 3))%>) by AFINSQ_2:53
.= (((1 * 1) + (1 * (10 |^ 1))) + (3 * (10 |^ 2))) + (1 * (10 |^ 3)) by AFINSQ_2:53
.= ((1 + (1 * 10)) + (3 * (10 |^ 2))) + (1 * (10 |^ 3)) by NEWTON:5
.= (11 + (3 * (10 * 10))) + (1 * (10 |^ 3)) by POLYEQ_5:1
.= (11 + (3 * (10 * 10))) + (1 * ((10 * 10) * 10)) by POLYEQ_5:2
.= 1311 ;
A2: dom <%1,1,3,1%> = 4 by AFINSQ_1:84
.= dom <%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2)),(1 * (10 |^ 3))%> by AFINSQ_1:84 ;

now :: thesis: for i being Nat st i in dom <%1,1,3,1%> holds
<%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2)),(1 * (10 |^ 3))%> . i = (<%1,1,3,1%> . i) * (10 |^ i)

let i be Nat; :: thesis: ( i in dom <%1,1,3,1%> implies <%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2)),(1 * (10 |^ 3))%> . i = (<%1,1,3,1%> . i) * (10 |^ i) )
assume i in dom <%1,1,3,1%> ; :: thesis: <%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2)),(1 * (10 |^ 3))%> . i = (<%1,1,3,1%> . i) * (10 |^ i)
then i in 4 by AFINSQ_1:84;
then i in {0,1,2,3} by CARD_1:52;
then ( i = 0 or i = 1 or i = 2 or i = 3 ) by ENUMSET1:def 2;
hence <%(1 * (10 |^ 0)),(1 * (10 |^ 1)),(3 * (10 |^ 2)),(1 * (10 |^ 3))%> . i = (<%1,1,3,1%> . i) * (10 |^ i) ; :: thesis: verum

end;

then A3: value (<%1,1,3,1%>,10) = 1311 by A1, A2, NUMERAL1:def 1;
(len <%1,1,3,1%>) - 1 = 4 - 1 by AFINSQ_1:84;
then A4: <%1,1,3,1%> . ((len <%1,1,3,1%>) - 1) <> 0 ;

now :: thesis: for i being Nat st i in dom <%1,1,3,1%> holds
( 0 <= <%1,1,3,1%> . i & <%1,1,3,1%> . i < 10 )

let i be Nat; :: thesis: ( i in dom <%1,1,3,1%> implies ( 0 <= <%1,1,3,1%> . i & <%1,1,3,1%> . i < 10 ) )
assume i in dom <%1,1,3,1%> ; :: thesis: ( 0 <= <%1,1,3,1%> . i & <%1,1,3,1%> . i < 10 )
then i in 4 by AFINSQ_1:84;
then i in {0,1,2,3} by CARD_1:52;
then ( i = 0 or i = 1 or i = 2 or i = 3 ) by ENUMSET1:def 2;
hence ( 0 <= <%1,1,3,1%> . i & <%1,1,3,1%> . i < 10 ) ; :: thesis: verum

end;

hence digits (1311,10) = <%1,1,3,1%> by A3, A4, NUMERAL1:def 2; :: thesis: verum