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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="graph_1.html#V3">oriented</a> <a href="graph_1.html#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>











<b>assume </b><a NAME="E1:98"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="graph_5.html#R5" target="_self">is_weight&gt;=0of</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="graph_5.html#R9" target="_self">islongestInShortestpath</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E2:98"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_1.html#T95">FINSEQ_1:95</a></i>;<br/>
<a NAME="E4:98"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E5:98"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E2:98"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_5.html#T38" target="_self">Th38</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E7:98"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E8:98"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E7:98"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E9:98"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E10:98"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E11:98"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E12:98"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E2:98"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_5.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E13:98"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E14:98"/>
<font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E15:98"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_5.html#E13:98"><i><font color="Green">E84</font></i></a>, <a class="ref" href="graph_5.html#T36" target="_self">Th36</a></i>;<br/>
<div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_1"/><i><font color="Green">E87</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_1_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:98_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:98_1_1_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:98"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><b>hence </b><a NAME="E3:98_1_1_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:98_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E15:98"><i><font color="Green">E85</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_1_2"/><i><font color="Green">E88</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:98_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:98_1_2_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:98"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E3:98_1_2_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:98_1_2_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:98_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:98"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E17:98"/><i><font color="Green">E87</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E9:98"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E18:98"/><i><font color="Green">E88</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/>
<a NAME="E19:98"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="graph_5.html#R6" target="_self">is_weight_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#T50" target="_self">Th50</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1"><b>suppose </b></a><a NAME="E1:98_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:98_2_1"/><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:98_2_1"/><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:98_2_1"><i><font color="Green">E90</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:98_2_1"/><i><font color="Green">E92</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="graph_5.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E8:98_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:98_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E9:98_2_1"/><i><font color="Green">E94</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E10:98_2_1"/><i><font color="Green">E95</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E11:98_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E12:98_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E13:98_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a></i>;<br/><a NAME="E14:98_2_1"/><b>then </b><i><font color="Green">E98</font></i>: 
( not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E17:98"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_5.html#E10:98_2_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="graph_5.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E15:98_2_1"/><i><font color="Green">E99</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:98_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E16:98_2_1"/><b>then </b><i><font color="Green">E100</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>;<br/><a NAME="E17:98_2_1"/><i><font color="Green">E101</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:98_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_5.html#E12:98_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_5.html#E15:98_2_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>;<br/><i><font color="Green">E102</font></i>: <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="graph_5.html#E17:98"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_5.html#T31" target="_self">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="graph_5.html#E14:98_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>
;<br/><a NAME="E19:98_2_1"/><i><font color="Green">E103</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:98"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="graph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E20:98_2_1"/>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:98"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_5.html#E18:98_2_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E21:98_2_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E22:98_2_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/><a NAME="E23:98_2_1"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E21:98_2_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="graph_5.html#E22:98_2_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_1_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font> or  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:98_2_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:98_2_1_2_1_1"/><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E11:98_2_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="graph_5.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E4:98_2_1_2_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E5:98_2_1_2_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_1_2_1_1_1_1"/>
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:98_2_1_2_1_1_1_1_1"/><i><font color="Green">E109</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:98_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E17:98"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_5.html#E18:98"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E5:98_2_1_2_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="graph_1.html#D4">GRAPH_1:def 4</a></i>;<br/><a NAME="E3:98_2_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E4:98_2_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E5:98_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E11:98_2_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_5.html#E17:98_2_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graph_5.html#E1:98_2_1_2_1_1_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:98_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E23:98_2_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E7:98_2_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E8:98_2_1_2_1_1_1_1_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E19:98"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_5.html#E3:98_2_1_2_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="graph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E9:98_2_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:98_2_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="graph_5.html#E7:98_2_1_2_1_1_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_1_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:98_2_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98_2_1_2_1_1_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:98_2_1_2_1_1_1_1_1_1_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:98"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><b>hence </b><a NAME="E3:98_2_1_2_1_1_1_1_1_1_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E2:98_2_1_2_1_1_1_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:98_2_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E15:98"><i><font color="Green">E85</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_1_2"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:98_2_1_2_1_1_1_1_1_2_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:98"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><b>hence </b><a NAME="E3:98_2_1_2_1_1_1_1_1_2_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98_2_1_2_1_1_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:98_2_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:98"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E109</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:98_2_1_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E4:98_2_1_2_1_1_1_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:98_2_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E12:98_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_5.html#E14:98_2_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graph_5.html#E16:98_2_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="graph_5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E6:98_2_1_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98_2_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="graph_5.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E8:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E112</font></i>: 
not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E17:98_2_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graph_5.html#E1:98_2_1_2_1_1_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:98_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:98"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_5.html#E18:98_2_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E10:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:98_2_1_2_1_1_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1_2_1_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E12:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E5:98_2_1_2_1_1_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="graph_5.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E13:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:98_2_1_2_1_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E14:98_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E15:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>21</sub></font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E3:98_2_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E12:98_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_5.html#E16:98_2_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="graph_5.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E17:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E18:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E19:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E15:98_2_1_2_1_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E20:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E15:98_2_1_2_1_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E21:98_2_1_2_1_1_1_1_2"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E17:98"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E17:98_2_1_2_1_1_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E22:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E18:98_2_1_2_1_1_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graph_5.html#E20:98_2_1_2_1_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E23:98_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E17:98"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_5.html#E6:98_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E8:98_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E19:98_2_1_2_1_1_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="graph_5.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E24:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/><a NAME="E25:98_2_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E13:98_2_1_2_1_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E26:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T44">XBOOLE_1:44</a></i>;<br/><a NAME="E27:98_2_1_2_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:98_2_1_2_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E10:98_2_1_2_1_1_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:98_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E29:98_2_1_2_1_1_1_1_2"/><b>then </b>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E26:98_2_1_2_1_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E30:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E31:98_2_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E24:98_2_1_2_1_1_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graph_5.html#E30:98_2_1_2_1_1_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E32:98_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E22:98_2_1_2_1_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E33:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E35:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:98_2_1_2_1_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E37:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E127</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E35:98_2_1_2_1_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="finseq_2.html#T25">FINSEQ_2:25</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>28</sub></font> as  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E37:98_2_1_2_1_1_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="graph_5.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E39:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E3:98_2_1_2_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graph_5.html#E37:98_2_1_2_1_1_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E40:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E129</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E11:98_2_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_5.html#E3:98_2_1_2_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graph_5.html#E37:98_2_1_2_1_1_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E41:98_2_1_2_1_1_1_1_2"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:98_2_1_2_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graph_5.html#E37:98_2_1_2_1_1_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E42:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E39:98_2_1_2_1_1_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="graph_5.html#E40:98_2_1_2_1_1_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E43:98_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>29</sub></font> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E44:98_2_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E37:98_2_1_2_1_1_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E45:98_2_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E23:98_2_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E46:98_2_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E42:98_2_1_2_1_1_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E47:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E48:98_2_1_2_1_1_1_1_2"/><i><font color="Green">E132</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:98"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_5.html#E37:98_2_1_2_1_1_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="graph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E49:98_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#T54" target="_self">Th54</a></i>;<br/><a NAME="E50:98_2_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E48:98_2_1_2_1_1_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E51:98_2_1_2_1_1_1_1_2"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E47:98_2_1_2_1_1_1_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E52:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:98_2_1_2_1_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E53:98_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:98"><i><font color="Green">E89</font></i></a>, <a class="ref" href="graph_5.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E54:98_2_1_2_1_1_1_1_2"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:98_2_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="graph_5.html#E52:98_2_1_2_1_1_1_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E55:98_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E33:98_2_1_2_1_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_2_2"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_2_2_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:98_2_1_2_1_1_1_1_2_2_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:98"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><b>hence </b><a NAME="E3:98_2_1_2_1_1_1_1_2_2_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E55:98_2_1_2_1_1_1_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:98_2_1_2_1_1_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E15:98"><i><font color="Green">E85</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_2_3"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:98_2_1_2_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E55:98_2_1_2_1_1_1_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2_1_1_1_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:98_2_1_2_1_1_1_1_2_3_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:98_2_1_2_1_1_1_1_2_3_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:98"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/><b>hence </b><a NAME="E3:98_2_1_2_1_1_1_1_2_3_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E2:98_2_1_2_1_1_1_1_2_3"><i><font color="Green">E135</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:98_2_1_2_1_1_1_1_2_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:98"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:98_2_1_2_1_2"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><a NAME="E2:98_2_1_2_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E3:98_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:98"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_5.html#E18:98_2_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E4:98_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E18:98_2_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="graph_5.html#E2:98_2_1_2_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/><a NAME="E5:98_2_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E6:98_2_1_2_1_2"/>
( (  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font> ) &amp; (  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E16:98"><i><font color="Green">E86</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2"><b>suppose </b></a><a NAME="E1:98_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:98_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:98_2_2"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E11:98"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:98_2_2"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E5:98_2_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_2_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:98_2_2_1"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K1" target="_self">vertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> &amp; not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="graph_5.html#E4:98_2_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:98_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
;<br/>
<a NAME="E4:98_2_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:98_2_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E11:98"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<font color="Maroon">c<sub>10</sub></font> = 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E9:98"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E3:98_2_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/>
<b>hence </b><a NAME="E7:98_2_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_5.html#E12:98"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_5.html#E2:98_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_5.html#E4:98_2_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_5.html#E5:98_2_2_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:98_2_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9:98"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E7:98_2_2"/>
( (  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font> ) &amp; (  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="graph_5.html#E1:98"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E16:98"><i><font color="Green">E86</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self">Def18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
