How do you prove this theorem?

How do you prove this theorem?

Let:
A = { 1, 3, 9, 27, 81, 243, 729 }
B = { 1, 3, 9, 27, 81, 243, 729 }
C = some combination of A (7 Choose k where k= 0 to 7)
D = some combination of B (7 choose k where k= 0 to 7)
G = {1 to 1066}
So,
for all x in G --> x = sum(C) - sum(D)
ie. All numbers in G can be represented from summing some combination from
A, minus the sum of some combination from B.
How would you prove that this is true in paper? I mean, I have tested it
and sum(C) - sum(D) does have a solution for all numbers in G. Other than
doing what I did (actually writing the code to solve it), how else would I
explain that this is true?