<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">W</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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> </span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<a NAME="E2:28"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:28"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="glib_004.html#D3" target="_self" title="GLIB_004:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
