deletions | additions       diff --git a/textbf_textit_Proof_of_property__1.tex b/textbf_textit_Proof_of_property__1.tex  index 8c45db5..a97b780 100644  --- a/textbf_textit_Proof_of_property__1.tex  +++ b/textbf_textit_Proof_of_property__1.tex 
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\end{itemize}  $$  {\textstyle  \frac{\partial}{\partial k}\left(x^{*}\left(\delta\right)+y^{*}\left(\delta\right)\right)  =-\frac{1-k}{\left(2\left(b-k\right)+\left(1-k\right)^{2}\right)^{2}}\sqrt{\frac{b-\delta}{b-k}}\frac{\phi\left(b,k\right)}{2}, k}\left(x^{*}\left(\delta\right)+y^{*}\left(\delta\right)\right){\textstyle =-\frac{1-k}{\left(2\left(b-k\right)+\left(1-k\right)^{2}\right)^{2}}\sqrt{\frac{b-\delta}{b-k}}\frac{\phi\left(b,k\right)}{2}},}  $$  $$  {\textstyle \frac{\partial}{\partial k}\pi\left(x^{*}\left(\delta\right),y^{*}\left(\delta\right)\right)}={\textstyle \frac{1-k}{\left(2\left(b-k\right)+\left(1-k\right)^{2}\right)^{2}}\sqrt{\frac{b-\delta}{b-k}}\frac{\phi\left(b,k\right)}{2\left(b-k\right)}},