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<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_1_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E4:12_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:12_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><b>hence </b><a NAME="E5:12_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_001.html#K26">.allDPaths()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#D38">GLIB_001:def 38</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_001.html#K26">.allDPaths()</a> </span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_1_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="glib_001.html#NM14">DPath</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E5:12_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:12_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E6:12_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_001.html#K26">.allDPaths()</a> </span>)</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="glib_001.html#M10">Element</a> of <font color="Maroon">c<sub>8</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="glib_003.html#K10">.cost()</a> ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="glib_001.html#M10">Element</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E9:12_1_1_2"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M10">Element</a> of <font color="Maroon">c<sub>8</sub></font> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2">GRAPH_5:sch 2</a>();<br/></i><a NAME="E10:12_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:12_1_1_2"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div><i><font color="Green">E12</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">c<sub>11</sub></font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:12_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:12_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="glib_001.html#M10">Element</a> of <font color="Maroon">c<sub>8</sub></font> ;<br/><a NAME="E4:12_1_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>12</sub></font>)
 <b>by </b><i><a class="txt" href="glib_004.html#E9:12_1_1_2"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><a NAME="E5:12_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E12:12_1_1_2"><i><font color="Green">E11</font></i></a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="glib_003.html#K10">.cost()</a> ;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="glib_003.html#K10">.cost()</a> ;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>thus </b><a NAME="E15:12_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E13:12_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="glib_004.html#E12:12_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="glib_004.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><a NAME="E16:12_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="glib_003.html#K10">.cost()</a> 
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E4:12_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">c<sub>1</sub></font> holds not <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


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