set d = <%1,5,5%>;
set e = <%(1 * (10 |^ 0)),(5 * (10 |^ 1)),(5 * (10 |^ 2))%>;
A1: Sum <%(1 * (10 |^ 0)),(5 * (10 |^ 1)),(5 * (10 |^ 2))%> = (Sum <%(1 * (10 |^ 0)),(5 * (10 |^ 1))%>) + (Sum <%(5 * (10 |^ 2))%>) by AFINSQ_2:55
.= ((1 * (10 |^ 0)) + (5 * (10 |^ 1))) + (Sum <%(5 * (10 |^ 2))%>) by AFINSQ_2:54
.= ((1 * 1) + (5 * (10 |^ 1))) + (5 * (10 |^ 2)) by AFINSQ_2:53
.= (1 + (5 * 10)) + (5 * (10 |^ 2)) by NEWTON:5
.= 51 + (5 * (10 * 10)) by POLYEQ_5:1
.= 551 ;
A2: dom <%1,5,5%> = 3 by AFINSQ_1:39
.= dom <%(1 * (10 |^ 0)),(5 * (10 |^ 1)),(5 * (10 |^ 2))%> by AFINSQ_1:39 ;

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

let i be Nat; :: thesis: ( i in dom <%1,5,5%> implies <%(1 * (10 |^ 0)),(5 * (10 |^ 1)),(5 * (10 |^ 2))%> . i = (<%1,5,5%> . i) * (10 |^ i) )
assume i in dom <%1,5,5%> ; :: thesis: <%(1 * (10 |^ 0)),(5 * (10 |^ 1)),(5 * (10 |^ 2))%> . i = (<%1,5,5%> . 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 <%(1 * (10 |^ 0)),(5 * (10 |^ 1)),(5 * (10 |^ 2))%> . i = (<%1,5,5%> . i) * (10 |^ i) ; :: thesis: verum

end;

then A3: value (<%1,5,5%>,10) = 551 by A1, A2, NUMERAL1:def 1;
(len <%1,5,5%>) - 1 = 3 - 1 by AFINSQ_1:39;
then A4: <%1,5,5%> . ((len <%1,5,5%>) - 1) <> 0 ;

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

let i be Nat; :: thesis: ( i in dom <%1,5,5%> implies ( 0 <= <%1,5,5%> . i & <%1,5,5%> . i < 10 ) )
assume i in dom <%1,5,5%> ; :: thesis: ( 0 <= <%1,5,5%> . i & <%1,5,5%> . 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 <= <%1,5,5%> . i & <%1,5,5%> . i < 10 ) ; :: thesis: verum

end;

hence digits (551,10) = <%1,5,5%> by A3, A4, NUMERAL1:def 2; :: thesis: verum