set d = <%5,8%>;
set e = <%(5 * (10 |^ 0)),(8 * (10 |^ 1))%>;
A1: Sum <%(5 * (10 |^ 0)),(8 * (10 |^ 1))%> = (5 * (10 |^ 0)) + (8 * (10 |^ 1)) by AFINSQ_2:54
.= (5 * 1) + (8 * (10 |^ 1)) by NEWTON:4
.= 5 + (8 * 10) by NEWTON:5 ;
A2: dom <%5,8%> = 2 by AFINSQ_1:38
.= dom <%(5 * (10 |^ 0)),(8 * (10 |^ 1))%> by AFINSQ_1:38 ;

now :: thesis: for i being Nat st i in dom <%5,8%> holds
<%(5 * (10 |^ 0)),(8 * (10 |^ 1))%> . i = (<%5,8%> . i) * (10 |^ i)

let i be Nat; :: thesis: ( i in dom <%5,8%> implies <%(5 * (10 |^ 0)),(8 * (10 |^ 1))%> . i = (<%5,8%> . i) * (10 |^ i) )
assume i in dom <%5,8%> ; :: thesis: <%(5 * (10 |^ 0)),(8 * (10 |^ 1))%> . i = (<%5,8%> . i) * (10 |^ i)
then i in 2 by AFINSQ_1:38;
then i in {0,1} by CARD_1:50;
then ( i = 0 or i = 1 ) by TARSKI:def 2;
hence <%(5 * (10 |^ 0)),(8 * (10 |^ 1))%> . i = (<%5,8%> . i) * (10 |^ i) ; :: thesis: verum

end;

then A3: value (<%5,8%>,10) = 85 by A1, A2, NUMERAL1:def 1;
(len <%5,8%>) - 1 = 2 - 1 by AFINSQ_1:38;
then A4: <%5,8%> . ((len <%5,8%>) - 1) <> 0 ;

now :: thesis: for i being Nat st i in dom <%5,8%> holds
( 0 <= <%5,8%> . i & <%5,8%> . i < 10 )

let i be Nat; :: thesis: ( i in dom <%5,8%> implies ( 0 <= <%5,8%> . i & <%5,8%> . i < 10 ) )
assume i in dom <%5,8%> ; :: thesis: ( 0 <= <%5,8%> . i & <%5,8%> . i < 10 )
then i in 2 by AFINSQ_1:38;
then i in {0,1} by CARD_1:50;
then ( i = 0 or i = 1 ) by TARSKI:def 2;
hence ( 0 <= <%5,8%> . i & <%5,8%> . i < 10 ) ; :: thesis: verum

end;

hence digits (85,10) = <%5,8%> by A3, A4, NUMERAL1:def 2; :: thesis: verum