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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</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="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">W1</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b2">W2</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> </span><br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</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:()"><i><font color="Red">::  thesis: </font></i></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">W1</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> &amp; <font color="Maroon" title="c3">W2</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="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> </span><br/><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">W1</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c3">W2</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  )</span><br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c2">W1</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c3">W2</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> </span><br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b1">W2</font> being  <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> holds <br/><font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">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#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/>
<a NAME="E3:27"/>
( <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive" title="b1">W1</font> being  <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> holds <br/><font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  ) )
 
<b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/>
<a NAME="E4:27"/><b>then </b>
( <font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  &amp; <font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  )
 
<b>by </b><i><a class="txt" href="#E2:27"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:27"/>
<font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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