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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be   <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">W</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E2:27"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">W2</font> being   <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:27"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c5">W2</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D3" target="_self" title="GLIB_004:def.3">Def3</a></i>;<br/>
<b>thus </b><a NAME="E5:27"/>
<font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:27"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="glib_004.html#E4:27"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="glib_004.html#T11" target="_self" title="GLIB_004:th.11">Th11</a></i>;<br/>


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