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<b>let </b><font color="Maroon" title="c4">IT1</font>, <font color="Maroon" title="c5">IT2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_2_1"/>
 ex <font color="Olive" title="b1">W</font> being  <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 ;<br/><b>given </b><font color="Maroon" title="c6">W1</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><a NAME="E3:12_1_2_1"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c6">W1</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  )
 ;<br/><b>given </b><font color="Maroon" title="c7">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><a NAME="E5:12_1_2_1"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c7">W2</font> <a href="glib_004.html#R2" title="GLIB_004:pred.2">is_mincost_DPath_from</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  )
 ;<br/><a NAME="E6:12_1_2_1"/><i><font color="Green" title="E11">A8</font></i>: 
( <font color="Maroon" title="c6">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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="c2">x</font>,<font color="Maroon" title="c3">y</font> holds <br/><font color="Maroon" title="c6">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="txt" href="glib_004.html#E3:12_1_2_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/><a NAME="E7:12_1_2_1"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c7">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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="c2">x</font>,<font color="Maroon" title="c3">y</font> holds <br/><font color="Maroon" title="c7">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="glib_004.html#E5:12_1_2_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/><a NAME="E8:12_1_2_1"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c5">IT2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">IT1</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:12_1_2_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="glib_004.html#E5:12_1_2_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="glib_004.html#E6:12_1_2_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/><a NAME="E9:12_1_2_1"/>
<font color="Maroon" title="c4">IT1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">IT2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:12_1_2_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="glib_004.html#E5:12_1_2_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="glib_004.html#E7:12_1_2_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E10:12_1_2_1"/>
<font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E8:12_1_2_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:12_1_2"/>
( ( for <font color="Olive" title="b1">W</font> being  <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G</font>  holds not <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> ) &amp; <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> )
 ;<br/>


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