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 by CARD_1:def 7;

then A8: dom k1 = Seg (n + 1) by FINSEQ_1:def 3;

hence k1 = k2 by A7, A9, FINSEQ_2:9; :: thesis: verum

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 by CARD_1:def 7;

then A8: dom k1 = Seg (n + 1) by FINSEQ_1:def 3;

A9: now :: thesis: for j being Nat st j in dom k1 holds

k1 . j = k2 . j

len k2 = n + 1
by CARD_1:def 7;k1 . j = k2 . j

let j be Nat; :: thesis: ( j in dom k1 implies k1 . j = k2 . j )

assume A10: j in dom k1 ; :: thesis: k1 . j = k2 . j

then k1 . j = DigA_SDSub (k1,j) by A8, Def5

.= SD2SDSubDigitS (x,j,k) by A5, A8, A10

.= DigA_SDSub (k2,j) by A6, A8, A10

.= k2 . j by A8, A10, Def5 ;

hence k1 . j = k2 . j ; :: thesis: verum

end;assume A10: j in dom k1 ; :: thesis: k1 . j = k2 . j

then k1 . j = DigA_SDSub (k1,j) by A8, Def5

.= SD2SDSubDigitS (x,j,k) by A5, A8, A10

.= DigA_SDSub (k2,j) by A6, A8, A10

.= k2 . j by A8, A10, Def5 ;

hence k1 . j = k2 . j ; :: thesis: verum

hence k1 = k2 by A7, A9, FINSEQ_2:9; :: thesis: verum