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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</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>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:59"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</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>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/> <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:59"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><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>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</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>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E51</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</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> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</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>1</sub></font> <b>by </b><i><a class="ref" href="graph_5.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E3:59_1"/>
<font color="Maroon">c<sub>10</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#E1:59_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E4:59_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="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>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>7</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:59_1"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:59"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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>1</sub></font>;<br/><b>let </b><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>1</sub></font>;<br/><b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:59_2_1"/><i><font color="Green">E54</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>12</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>13</sub></font> = <font color="Maroon">c<sub>11</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>1</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_5.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E5:59_2_1"/><i><font color="Green">E55</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:59_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="graph_5.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E8:59_2_1"/><i><font color="Green">E57</font></i>: 
<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>8</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</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>13</sub></font></span>)</span></span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E6:59_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="graph_5.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E10:59_2_1"/>
<font color="Maroon">c<sub>13</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#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E11:59_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E12:59_2_1"/><b>then </b>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>13</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#E1:59_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:59_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E15:59_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="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>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E14:59_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</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>1</sub></font><b> such that </b><br/><a NAME="E17:59_2_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E18:59_2_1"/><i><font color="Green">E60</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:59_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E20:59_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E62</font></i>: the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>17</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E5:59_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_5.html#E17:59_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>
;<br/><a NAME="E22:59_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><a NAME="E23:59_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><a NAME="E24:59_2_1"/><i><font color="Green">E65</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>17</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E18:59_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="graph_5.html#E21:59_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> ) or for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> or  not the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_1"><b>suppose </b></a><a NAME="E1:59_2_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:59_2_1_2_1"/><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:59_2_1_2_1"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_2_1"><i><font color="Green">E66</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>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:59_2_1_2_1"/><i><font color="Green">E68</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>18</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E5:59_2_1_2_1"><i><font color="Green">E67</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>22</sub></font> = <font color="Maroon">c<sub>20</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>1</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E7:59_2_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="graph_5.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E9:59_2_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:59_2_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><a NAME="E10:59_2_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E17:59_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:59_2_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_2_1"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E12:59_2_1_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>22</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#E7:59_2_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E13:59_2_1_2_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E22:59_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E10:59_2_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:59_2_1_2_1"/><i><font color="Green">E72</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:59_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_5.html#E17:59_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E15:59_2_1_2_1"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</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>22</sub></font></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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E19:59_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E16:59_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="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>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>8</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E12:59_2_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_5.html#E13:59_2_1_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_5.html#E14:59_2_1_2_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E17:59_2_1_2_1"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_2"><b>suppose </b></a><a NAME="E1:59_2_1_2_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> or  not the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>12</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>1</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_5.html#T14" target="_self">Th14</a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_2_2_1_1"/>
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>17</sub></font></span>)</span></span>)</span> or the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>17</sub></font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_2_1_2_2_1_1_1"/><i><font color="Green">E67</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>17</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font>;<br/><a NAME="E2:59_2_1_2_2_1_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E17:59_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:59_2_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font> is  <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>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E19:59_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_5.html#E21:59_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_5.html#E1:59_2_1_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_5.html#E1:59_2_1_2_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="graph_5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:59_2_1_2_2_1_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:59_2_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</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="E6:59_2_1_2_2_1_1_1"/><i><font color="Green">E72</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:59_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_5.html#E17:59_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_5.html#E2:59_2_1_2_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E7:59_2_1_2_2_1_1_1"/><i><font color="Green">E73</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E19:59_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_5.html#E4:59_2_1_2_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E8:59_2_1_2_2_1_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E9:59_2_1_2_2_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E17:59_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_5.html#E22:59_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:59_2_1_2_2_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E23:59_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_5.html#E8:59_2_1_2_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/><a NAME="E11:59_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="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>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>8</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1_2_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_5.html#E5:59_2_1_2_2_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_5.html#E6:59_2_1_2_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_5.html#E7:59_2_1_2_2_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/><b>hence </b><a NAME="E12:59_2_1_2_2_1_1_1"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_2_1_2_2_1_1_2"/><i><font color="Green">E67</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>17</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</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>1</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:59_2_1_2_2_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>19</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#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:59_2_1_2_2_1_1_2"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</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>19</sub></font></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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:59_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E5:59_2_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="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>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></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>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>8</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E8:59_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_5.html#E17:59_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_5.html#E23:59_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_5.html#E24:59_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E1:59_2_1_2_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_5.html#E3:59_2_1_2_2_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E6:59_2_1_2_2_1_1_2"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<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><b>end;</b></div>
<b>hence </b><a NAME="E3:59_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:59"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="graph_5.html#E2:59"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_5.html#E3:59"><i><font color="Green">E51</font></i></a>);<br/></i>
<a NAME="E5:59"/>
<font color="Maroon">c<sub>4</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#E1:59"><i><font color="Green">E49</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:59"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</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="E7:59"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>hence </b><a NAME="E8:59"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>4</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#E1:59"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_5.html#E4:59"><i><font color="Green">E52</font></i></a></i>;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:59"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E11:59"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K4" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:59"><i><font color="Green">E49</font></i></a>, <a class="ref" href="graph_5.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E12:59"/>
 <a href="graph_5.html#K4" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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#E10:59"><i><font color="Green">E53</font></i></a></i>;<br/>


</div>
