deffunc H₁( Nat) -> Element of k -SD_Sub = SD2SDSubDigitS (x,$1,k);
consider z being FinSequence of k -SD_Sub such that
A1: len z = n + 1 and
A2: for j being Nat st j in dom z holds
z . j = H₁(j) from FINSEQ_2:sch 1();
A3: dom z = Seg (n + 1) by A1, FINSEQ_1:def 3;
z is Element of (n + 1) -tuples_on (k -SD_Sub) by A1, FINSEQ_2:92;
then reconsider z = z as Tuple of n + 1,k -SD_Sub ;
take z ; :: thesis: for i being Nat st i in Seg (n + 1) holds
DigA_SDSub (z,i) = SD2SDSubDigitS (x,i,k)
let i be Nat; :: thesis: ( i in Seg (n + 1) implies DigA_SDSub (z,i) = SD2SDSubDigitS (x,i,k) )
assume A4: i in Seg (n + 1) ; :: thesis: DigA_SDSub (z,i) = SD2SDSubDigitS (x,i,k)
hence DigA_SDSub (z,i) = z . i by Def5
.= SD2SDSubDigitS (x,i,k) by A2, A3, A4 ;
:: thesis: verum