set d = <%0,0,5%>;
set e = <%(0 * (10 |^ 0)),(0 * (10 |^ 1)),(5 * (10 |^ 2))%>;
A1: Sum <%(0 * (10 |^ 0)),(0 * (10 |^ 1)),(5 * (10 |^ 2))%> = (Sum <%(0 * (10 |^ 0)),(0 * (10 |^ 1))%>) + (Sum <%(5 * (10 |^ 2))%>) by AFINSQ_2:55
.= ((0 * (10 |^ 0)) + (0 * (10 |^ 1))) + (Sum <%(5 * (10 |^ 2))%>) by AFINSQ_2:54
.= ((0 * 1) + (0 * (10 |^ 1))) + (5 * (10 |^ 2)) by AFINSQ_2:53
.= 0 + (5 * (10 * 10)) by POLYEQ_5:1
.= 500 ;
A2: dom <%0,0,5%> = 3 by AFINSQ_1:39
.= dom <%(0 * (10 |^ 0)),(0 * (10 |^ 1)),(5 * (10 |^ 2))%> by AFINSQ_1:39 ;

now :: thesis: for i being Nat st i in dom <%0,0,5%> holds
<%(0 * (10 |^ 0)),(0 * (10 |^ 1)),(5 * (10 |^ 2))%> . i = (<%0,0,5%> . i) * (10 |^ i)

let i be Nat; :: thesis: ( i in dom <%0,0,5%> implies <%(0 * (10 |^ 0)),(0 * (10 |^ 1)),(5 * (10 |^ 2))%> . i = (<%0,0,5%> . i) * (10 |^ i) )
assume i in dom <%0,0,5%> ; :: thesis: <%(0 * (10 |^ 0)),(0 * (10 |^ 1)),(5 * (10 |^ 2))%> . i = (<%0,0,5%> . i) * (10 |^ i)
then i in 3 by AFINSQ_1:39;
then i in {0,1,2} by CARD_1:51;
then ( i = 0 or i = 1 or i = 2 ) by ENUMSET1:def 1;
hence <%(0 * (10 |^ 0)),(0 * (10 |^ 1)),(5 * (10 |^ 2))%> . i = (<%0,0,5%> . i) * (10 |^ i) ; :: thesis: verum

end;

then A3: value (<%0,0,5%>,10) = 500 by A1, A2, NUMERAL1:def 1;
(len <%0,0,5%>) - 1 = 3 - 1 by AFINSQ_1:39;
then A4: <%0,0,5%> . ((len <%0,0,5%>) - 1) <> 0 ;

now :: thesis: for i being Nat st i in dom <%0,0,5%> holds
( 0 <= <%0,0,5%> . i & <%0,0,5%> . i < 10 )

let i be Nat; :: thesis: ( i in dom <%0,0,5%> implies ( 0 <= <%0,0,5%> . i & <%0,0,5%> . i < 10 ) )
assume i in dom <%0,0,5%> ; :: thesis: ( 0 <= <%0,0,5%> . i & <%0,0,5%> . i < 10 )
then i in 3 by AFINSQ_1:39;
then i in {0,1,2} by CARD_1:51;
then ( i = 0 or i = 1 or i = 2 ) by ENUMSET1:def 1;
hence ( 0 <= <%0,0,5%> . i & <%0,0,5%> . i < 10 ) ; :: thesis: verum

end;

hence digits (500,10) = <%0,0,5%> by A3, A4, NUMERAL1:def 2; :: thesis: verum