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<b>let </b><font color="Maroon">G</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">W1</font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>, <font color="Maroon">W2</font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">W1</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">W2</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; ( for <font color="Olive">W2</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> holds <br/><font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">W2</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 
<b>by </b><i/>;<br/>
<a NAME="E3:26"/>
( <font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; ( for <font color="Olive">W1</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> holds <br/><font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">W1</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:26"/><b>then </b>
( <font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  &amp; <font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:26"/>
<font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</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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