Question of proving the opening of Pedersen Commitment
With this zero knowledge proof, the prover attempts to prove that he knows an opening of the commitment.

The obvious way to approach this is to show that, if the prover is able to complete the protocol for two different challenges, he is able to construct an opening of the commitment (and hence he actually knows what he is attempting to prove knowledge of).

How we show this is straight-forward:


Suppose the prover knows a value $y$ for which he can correctly answer two distinct challenge values $e, e'$.  That is, if given the challenge $e$, he can answer with $z_1, z_2$ such that $g^{z_1}h^{z_2} = yc^e$, and if given the challenge $e'$, he can answer with $z'_1, z'_2$ such that $g^{z'_1}h^{z'_2} = yc^{e'}$
Then, given these values, the prover then knows:

$$g^{(z_1-z'_1)(e - e')^{-1}}h^{(z_2-z'_2)(e - e')^{-1}} = c$$


That is, with valid answers to those two challenges, he can deduce an opening of the commitment.

Hence, someone who doesn't know a valid opening to the commitment can provide a valid answer at most one challenge to any $y$ value.