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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>, <font color="Maroon">c<sub>3</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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green">E15</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_003.html#K10">.cost()</a>  ) )
 
<b>by </b><i><a class="ref" href="glib_004.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:24"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_003.html#K10">.cost()</a>  ) )
 
<b>by </b><i><a class="txt" href="glib_004.html#E1:24"><i><font color="Green">E15</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:24"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_003.html#K10">.cost()</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</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#E2:24"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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