let k1, k2 be Tuple of n,k -SD ; :: thesis: ( ( for i being Nat st i in Seg n holds
DigA (k1,i) = Add (x,y,i,k) ) & ( for i being Nat st i in Seg n holds
DigA (k2,i) = Add (x,y,i,k) ) implies k1 = k2 )
assume that
A5: for i being Nat st i in Seg n holds
DigA (k1,i) = Add (x,y,i,k) and
A6: for i being Nat st i in Seg n holds
DigA (k2,i) = Add (x,y,i,k) ; :: thesis: k1 = k2
A7: len k1 = n by CARD_1:def 7;
then A8: dom k1 = Seg n by FINSEQ_1:def 3;
A9: now :: thesis: for j being Nat st j in dom k1 holds
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 (k1,j) by A8, Def3
.= Add (x,y,j,k) by A5, A8, A10
.= DigA (k2,j) by A6, A8, A10
.= k2 . j by A8, A10, Def3 ;
hence k1 . j = k2 . j ; :: thesis: verum
end;
len k2 = n by CARD_1:def 7;
hence k1 = k2 by A7, A9, FINSEQ_2:9; :: thesis: verum