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Proof of Lemma 1 paper " Provably Secure Partially Blind Signatures Masayuki ABE and Tatsuaki OKAMOTO"

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I'm troubled by a system of equations presented in the paper "Provably Secure Partially Blind Signatures" Masayuki ABE and Tatsuaki OKAMOTO.

Proof part 1

proof part 2

In lemma 1 the authors define $t_2=w_j-c_i$ however $c_i$ following the signature protocol creates a feed back loop as $t_2$ is defined and used as input in a secure hash function. Signature protocol

Why is it true that the equations always have a solution?

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https://www.iacr.org/archive/crypto2000/18800272/18800272.pdf
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