View source for Claim:The base b representation is unique

As usual, in order to justify calling the [[base b representation]] of <math>a</math> "the" [[base b representation]], we need to prove that it is unique.  In fact, it is not necessarily unique if we allow leading 0's (for example, <math>(0123)_{10} = (123)_{10}</math>), but if we rule them out, then we have the following fact:

{{Claim|1=The base b representation is unique|2=
[[For all]] <math>k</math>, [[if]] <math>[[base|(d_kd_{k-1}\dots{}d_1d_0)_b]] = [[base|(d_k'd_{k-1}'\dots{}d_1'd_0')_b]]</math> and if <math>d_k \neq 0</math> and if <math>d_k' \neq 0</math> then <math>[[sequence notation|(d_i)]] = [[sequence notation|(d_i')]]</math>.
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