let k1, k2 be Tuple of (n + 1),(k -SD_Sub ); :: thesis: ( ( for i being Nat st i in Seg (n + 1) holds
DigA_SDSub k1,i = SD2SDSubDigitS x,i,k ) & ( for i being Nat st i in Seg (n + 1) holds
DigA_SDSub k2,i = SD2SDSubDigitS x,i,k ) implies k1 = k2 )
assume that
A5: for i being Nat st i in Seg (n + 1) holds
DigA_SDSub k1,i = SD2SDSubDigitS x,i,k and
A6: for i being Nat st i in Seg (n + 1) holds
DigA_SDSub k2,i = SD2SDSubDigitS x,i,k ; :: thesis: k1 = k2
A7: ( len k1 = n + 1 & len k2 = n + 1 ) by FINSEQ_1:def 18;
X: dom k1 = Seg (n + 1) by A7, FINSEQ_1:def 3;

now

let j be Nat; :: thesis: ( j in dom k1 implies k1 . j = k2 . j )
assume A8: j in dom k1 ; :: thesis: k1 . j = k2 . j
then k1 . j = DigA_SDSub k1,j by Def5, X
.= SD2SDSubDigitS x,j,k by A5, A8, X
.= DigA_SDSub k2,j by A6, A8, X
.= k2 . j by A8, Def5, X ;
hence k1 . j = k2 . j ; :: thesis: verum

end;

hence k1 = k2 by A7, FINSEQ_2:10; :: thesis: verum