## Appendix B  
**Proof of Proposition 1. ** The optimal bale diameter \(D_{b}^{*}\) is near-equal to the optimal bale height \(H_{b}^{*}\) with order \(\varepsilon_{0}\) if and only if Equations (1) and (5) are satisfied, simultaneously. For \(D_{b}^{*}\) given by Equation (5) the optimality condition, Equation (1), takes the form  
\[\frac{2\pi p_{k}}{\Omega\big{(}k_{f}\big{)}}\frac{4(1+\varepsilon_{0})V_{00}} {\pi}+\frac{\pi p_{k}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left(\sqrt{ \frac{4(1+\varepsilon_{0})V_{00}}{\pi}}\right)^{2}-4V_{0b}\left[\frac{p_{b}}{ 1-k_{fb}}+\frac{p_{u}}{\Omega\big{(}k_{f}\big{)}}\right]=0\]  
and is equivalent to the next equation  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]  
which can be rewritten as follows  
\[\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left(\sqrt[3]{1+ \varepsilon_{0}}\right)^{2}\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[}\frac{p_{b}}{ 1-k_{fb}}-\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}-\frac{2p_{u}}{\Omega\Big{(} k_{f}\Big{)}}\varepsilon_{0}\Bigg{]}. \tag{10}\]  
To show the Proposition two cases must be considered separately:  
1. the wrapping parameters and order \(\varepsilon_{0}\) are such that \[\frac{p_{b}}{1-k_{fb}}\
eq\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}+\frac{2p_{u }}{\Omega\Big{(}k_{f}\Big{)}}\varepsilon_{0},\] (11)
2. for the wrapping parameters and \(\varepsilon_{0}\) the following equality holds \[\frac{p_{b}}{1-k_{fb}}=\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}+\frac{2p_{u}}{ \Omega\Big{(}k_{f}\Big{)}}\varepsilon_{0}.\] (12)  
Provided that inequality from Equation (11) is satisfied, we have  
\[\frac{\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left(\sqrt[3]{1 +\varepsilon_{0}}\right)^{2}}{\frac{p_{b}}{1-k_{fb}}-\frac{p_{u}}{\Omega \big{(}k_{f}\big{)}}-\frac{2p_{u}}{\Omega\big{(}k_{f}\big{)}}\varepsilon_{0}}= \sqrt[3]{\frac{4V_{00}}{\pi}}, \tag{13}\]  
whence Equation (7) follows. The optimal bale diameter and height are described by Equations (5) and (6) and for \(D_{b}^{*}\) and \(H_{b}^{*}\) Equations (1) and (5) are satisfied; case (i) is proved. In case (ii), when Equation (12) holds, Equation (10) is satisfied if and only if, simultaneously,  
\[2\delta-b_{fr}k_{fb}=0.\]Then, Equation (1) takes especially simple form  
\[\frac{2\pi p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\big{(}D_{b}^{*}\big{)}^{3}-4V_{b0} \Bigg{[}\frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]  
whence the optimal bale diameter  
\[D_{b}^{*}=\sqrt[3]{\frac{2V_{b0}}{\pi}\Bigg{[}\frac{p_{b}\Omega\Big{(}k_{f} \Big{)}}{p_{u}\Big{(}1-k_{fb}\Big{)}}+1\Bigg{]}}.\] (A9)  
Simultaneously, from Equation (A7) it follows that  
\[\frac{p_{b}\Omega\Big{(}k_{f}\Big{)}}{p_{u}\Big{(}1-k_{fb}\Big{)}}=1+2\varepsilon _{0}.\]  
Thus Equations (A9) and (5) are identical and, by Equation (4), formula from Equation (6) follows for any \(V_{b0}\), which completes the proof of case (ii) since Equations (A7) and (8) are equivalent. **Proof of Proposition 2. ** By Proposition 1 the optimal bale dimensions are near-equal with order \(\varepsilon_{0}\) if:  
* in case (i) inequality expressed by Equation (13) is satisfied,
* in case (ii) \[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]0 (A10)  
and, simultaneously, Equation (9) holds. In the second case any change of \(\varepsilon_{0}\) to \(\varepsilon<\varepsilon_{0}\) results in a loss of equality in Equation (A10).