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\begin{document}
  	\begin{align*}
  	Pr_{(g_{1},g_{2})}  & = \dfrac{\mid \lbrace  [ (x_{1},x_{2}) , (y_{1},y_{2}) ]\in (G_{1}\times  G_{2} )^{2}  : [ (x_{1},x_{2}) , (y_{1},y_{2}) ]= (g_{1},g_{2}) \rbrace \mid}{ \mid G_{1} \times  G_{2}\mid^{2} } \\
  	& = \dfrac{\mid \lbrace  [ (x_{1},x_{2}) , (y_{1},y_{2}) ]\in (G_{1}\times  G_{2} )^{2}  : ( [x_{1},y_{1}] , [x_{2},y_{2}] ) = (g_{1},g_{2}) \rbrace \mid}{ \mid G_{1} \mid^{2} \times \mid G_{2}\mid^{2} }\\
  	&  =\dfrac{\mid \lbrace  (x_{1},y_{1})\in G_{1}^{2}  : \left[ x_{1},y_{1} \left]=g_{1}    \rbrace \mid}{  \mid G_{1} \mid^{2} } \times \dfrac{\mid \lbrace  (x_{2},y_{2})\in G_{2}^{2}  : \left[ x_{2},y_{2} \left]=g_{2}    \rbrace \mid}{  \mid G_{2} \mid^{2} } \\
  	&  = Pr_{(g_{1})}(G_{1}) \times Pr_{(g_{2})}(G_{1})  \\
  	\end{align*}	
	
		\begin{align*}
		g_{1} \in \left\langle  \hat{X},\Phi(G)   \right\rangle  & \Rightarrow x^{\alpha_{s}}_{s} \in \left\langle  \hat{X},\Phi(G)   \right\rangle   
		\Rightarrow x_{s} \in \left\langle  \hat{X},\Phi(G)   \right\rangle    \quad  (6) \\
		(5), (6) & \Rightarrow  \left\langle   X \right\rangle  \subseteq \left\langle  \hat{X},\Phi(G)   \right\rangle  
		\Rightarrow G = \left\langle  \hat{X},\Phi(G)  \right\rangle  \\
		& \Rightarrow  G = \left\langle  \hat{X}  \left\rangle  .
		\end{align*}
	
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