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<b>set </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">W</font> where <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font> : <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> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">W</font> where <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font> : <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> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">W</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="E3:12_1_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c6">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/><a NAME="E4:12_1_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:12_1_1_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:12_1_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K26" title="GLIB_001:func.26">.allDPaths()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#D38" title="GLIB_001:def.38">GLIB_001:def 38</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">W</font> where <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font> : <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> </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K26" title="GLIB_001:func.26">.allDPaths()</a> </span>)</span> <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_2"/>
 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>then consider </b><font color="Maroon" title="c6">W</font> being   <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:12_1_1_2"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c6">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>consider </b><font color="Maroon" title="c7">P</font> being   <a href="glib_001.html#NM14" title="GLIB_001:NM.14">DPath</a> of <font color="Maroon" title="c6">W</font>;<br/><a NAME="E5:12_1_1_2"/>
<font color="Maroon" title="c7">P</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>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:12_1_1_2"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><a NAME="E6:12_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c5">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K26" title="GLIB_001:func.26">.allDPaths()</a> </span>)</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="glib_001.html#M10" title="GLIB_001:mode.10">Element</a> of <font color="Maroon" title="c8">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = $1 <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> ;<br/><b>consider </b><font color="Maroon" title="c9">W1</font> being    <a href="glib_001.html#M10" title="GLIB_001:mode.10">Element</a> of <font color="Maroon" title="c8">X</font><b> such that </b><br/><a NAME="E9:12_1_1_2"/><i><font color="Green" title="E10">A3</font></i>: 
for <font color="Olive" title="b1">W2</font> being   <a href="glib_001.html#M10" title="GLIB_001:mode.10">Element</a> of <font color="Maroon" title="c8">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">W1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">W2</font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></i><a NAME="E10:12_1_1_2"/>
<font color="Maroon" title="c9">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">WA</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="E12:12_1_1_2"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c10">WA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W1</font> &amp; <font color="Maroon" title="c10">WA</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/><div><i><font color="Green" title="E12">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">WB</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>assume </b><a NAME="E1:12_1_1_2_1"/>
<font color="Maroon" title="c11">WB</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/><a NAME="E2:12_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c11">WB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">WB'</font> = <font color="Maroon" title="c11">WB</font> as    <a href="glib_001.html#M10" title="GLIB_001:mode.10">Element</a> of <font color="Maroon" title="c8">X</font> ;<br/><a NAME="E4:12_1_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">W1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">WB'</font>)
 <b>by </b><i><a class="txt" href="glib_004.html#E9:12_1_1_2"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:12_1_1_2_1"/>
<font color="Maroon" title="c10">WA</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="c11">WB</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#E12:12_1_1_2"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c11">WA</font> = <font color="Maroon" title="c10">WA</font> as   <a href="glib_001.html#NM8" title="GLIB_001:NM.8">DPath</a> of <font color="Maroon" title="c1">G</font> ;<br/><b>set </b><font color="Maroon" title="c12">IT</font> = <font color="Maroon" title="c11">WA</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> ;<br/><b>take </b><font color="Maroon" title="c13">IT</font> = <font color="Maroon" title="c11">WA</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> ;<br/><b>take </b><font color="Maroon" title="c14">WA</font> = <font color="Maroon" title="c11">WA</font>;<br/><b>thus </b><a NAME="E15:12_1_1_2"/>
<font color="Maroon" title="c14">WA</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>
 <b>by </b><i><a class="txt" href="glib_004.html#E12:12_1_1_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="glib_004.html#E13:12_1_1_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self" title="GLIB_004:def.2">Def2</a></i>;<br/><b>thus </b><a NAME="E16:12_1_1_2"/>
<font color="Maroon" title="c13">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">WA</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> 
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E4:12_1_1"/>
( ( 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> ) implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>


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