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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:25"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="glib_004.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:25"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="glib_003.html#K10">.cost()</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><a NAME="E5:25"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:25"><i><font color="Green">E16</font></i></a>, <a class="txt" href="glib_004.html#E4:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="glib_004.html#T11" target="_self">Th11</a></i>;<br/>


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