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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</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>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:60"/><i><font color="Green">E50</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</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>2</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>2</sub></font> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>1</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>6</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:60"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </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>2</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>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:60_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:60_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:60_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</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="E5:60_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:60_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</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>2</sub></font> <b>by </b><i><a class="ref" href="graph_5.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E7:60_1"/>
<font color="Maroon">c<sub>11</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>2</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>2</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>2</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>2</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>2</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#E3:60_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:60_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>
<a NAME="E3:60"/><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> 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="60_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:60_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</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>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</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>2</sub></font>;<br/><b>assume </b><a NAME="E1:60_2_1"/><i><font color="Green">E54</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:60_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="graph_4.html#T10">GRAPH_4:10</a></i>;<br/><a NAME="E4:60_2_1"/><i><font color="Green">E56</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</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>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 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>9</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/><a NAME="E5:60_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E6:60_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>9</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="E7:60_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</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="E8:60_2_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <font color="Maroon">c<sub>9</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>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E9:60_2_1"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E4:60_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E10:60_2_1"/>
<font color="Maroon">c<sub>9</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>9</sub></font></span>)</span> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</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>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_5.html#E7:60_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/><a NAME="E11:60_2_1"/><b>then </b>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</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>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/><a NAME="E12:60_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</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>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:60_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_5.html#E9:60_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2_1_1"/>
( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_1"><b>suppose </b></a><a NAME="E1:60_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:60_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="graph_4.html#V1">Simple</a>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="graph_4.html#D7">GRAPH_4:def 7</a></i>;<br/><a NAME="E3:60_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</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>2</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</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>9</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E4:60_2_1_1_1"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>9</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="60_2_1_1_2"><b>suppose </b></a><a NAME="E1:60_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:60_2_1_1_2"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; (  not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> ) )
 ;<br/><a NAME="E4:60_2_1_1_2"/><i><font color="Green">E64</font></i>: 
<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="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:60_2_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:60_2_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:60_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:60_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:60_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:60_2_1_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E6:60_2_1_1_2_1_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <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>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E5:60_2_1_1_2_1_1_1"><i><font color="Green">E68</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:60_2_1_1_2_1_1_1"/><i><font color="Green">E70</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>15</sub></font> &amp;  <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>16</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T25">FINSEQ_2:25</a></i>;<br/><a NAME="E9:60_2_1_1_2_1_1_1"/><i><font color="Green">E71</font></i>: 
<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>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:60_2_1_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_5.html#E7:60_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/><a NAME="E11:60_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E1:60_2_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_5.html#E3:60_2_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/><b>then </b><i><font color="Green">E73</font></i>: the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E1:60_2_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E9:60_2_1_1_2_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="graph_5.html#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>
;<br/><a NAME="E13:60_2_1_1_2_1_1_1"/><i><font color="Green">E74</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E14:60_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E8:60_2_1_1_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="E15:60_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>18</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>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</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>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_5.html#E13:60_2_1_1_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E16:60_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>18</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</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>2</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_5.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E18:60_2_1_1_2_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#E13:60_2_1_1_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E19:60_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:60_2_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_5.html#E14:60_2_1_1_2_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_5.html#E16:60_2_1_1_2_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E20:60_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></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>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E21:60_2_1_1_2_1_1_1"/><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>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="graph_5.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E22:60_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E19:60_2_1_1_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E23:60_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E6:60_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E1:60_2_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E24:60_2_1_1_2_1_1_1"/><b>then </b>
 <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>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E25:60_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:60_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E27:60_2_1_1_2_1_1_1"/><b>then </b>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <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#E1:60_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_5.html#E22:60_2_1_1_2_1_1_1"><i><font color="Green">E78</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E29:60_2_1_1_2_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E30:60_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>21</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>2</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>2</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>10</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</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>11</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>20</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E12:60_2_1_1_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_5.html#E14:60_2_1_1_2_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_5.html#E16:60_2_1_1_2_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E29:60_2_1_1_2_1_1_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</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>2</sub></font><b> such that </b><br/><a NAME="E32:60_2_1_1_2_1_1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <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>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>10</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>11</sub></font> &amp;  <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>20</sub></font> )
 ;<br/><a NAME="E33:60_2_1_1_2_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><a NAME="E34:60_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E81</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>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E32:60_2_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E35:60_2_1_1_2_1_1_1"/>
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_5.html#E32:60_2_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E36:60_2_1_1_2_1_1_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>2</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</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>9</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E32:60_2_1_1_2_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_5.html#E34:60_2_1_1_2_1_1_1"><i><font color="Green">E81</font></i></a></i>;<br/><b>hence </b><a NAME="E37:60_2_1_1_2_1_1_1"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>9</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="60_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:60_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:60_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:60_2_1_1_2_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E5:60_2_1_1_2_1_1_2"/><i><font color="Green">E68</font></i>: 
<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>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:60_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:60_2_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_5.html#E4:60_2_1_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><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="E8:60_2_1_1_2_1_1_2"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <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/><b>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="E10:60_2_1_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</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> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>9</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="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_2.html#T25">FINSEQ_2:25</a></i>;<br/><a NAME="E11:60_2_1_1_2_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E2:60_2_1_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_5.html#E4:60_2_1_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:60_2_1_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><a NAME="E13:60_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_5.html#E11:60_2_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E14:60_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1</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#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E15:60_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_5.html#E6:60_2_1_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_5.html#E11:60_2_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/><b>then </b><i><font color="Green">E75</font></i>: the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="graph_5.html#E4:60_2_1_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E5:60_2_1_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/><a NAME="E17:60_2_1_1_2_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>18</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>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_5.html#E11:60_2_1_1_2_1_1_2"><i><font color="Green">E72</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>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_1.html#M2">Chain</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</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="60_2_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2_1_1_2_1_1_2_2_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:60_2_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_2_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_5.html#E12:60_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E16:60_2_1_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_5.html#E17:60_2_1_1_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:60_2_1_1_2_1_1_2_2_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#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_5.html#E11:60_2_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:60_2_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E14:60_2_1_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_5.html#E2:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E5:60_2_1_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" target="_self">vertices</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></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>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E6:60_2_1_1_2_1_1_2_2_1_1"/><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>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="graph_5.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E7:60_2_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E8:60_2_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E6:60_2_1_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:60_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:60_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <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#E1:60_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_5.html#E7:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:60_2_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E13:60_2_1_1_2_1_1_2_2_1_1"/>
<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>2</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>2</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>10</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</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>11</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>20</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E14:60_2_1_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_5.html#E2:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_5.html#E12:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E80</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</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>2</sub></font><b> such that </b><br/><a NAME="E15:60_2_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</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>23</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><a NAME="E16:60_2_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:60_2_1_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_5.html#E15:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:60_2_1_1_2_1_1_2_2_1_1"/>
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</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>23</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_5.html#E15:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E18:60_2_1_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</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>2</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</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>9</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E15:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_5.html#E16:60_2_1_1_2_1_1_2_2_1_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>hence </b><a NAME="E19:60_2_1_1_2_1_1_2_2_1_1"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>9</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="60_2_1_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:60_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:60_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><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="E5:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E6:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <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>22</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E5:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E78</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E80</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>24</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>25</sub></font> = <font color="Maroon">c<sub>24</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>2</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="graph_5.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E10:60_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_5.html#E4:60_2_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_5.html#E3:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/><b>then </b><i><font color="Green">E81</font></i>: the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>
<br/>.= 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</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#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_5.html#E4:60_2_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_5.html#E16:60_2_1_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_5.html#E17:60_2_1_1_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/><a NAME="E12:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E82</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E13:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</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>2</sub></font> <b>by </b><i><a class="txt" href="graph_5.html#E11:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="graph_5.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E15:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E83</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> 1</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#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_5.html#E11:60_2_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_5.html#E14:60_2_1_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma9</font></i></a></i>;<br/><font color="Maroon">c<sub>26</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>26</sub></font></span>)</span> = 
<font color="Maroon">c<sub>26</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>20</sub></font></span>)</span> <a href="nat_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>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</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>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E12:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</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>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E6:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_5.html#E12:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma10</font></i></a></i>
;<br/><a NAME="E17:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</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>26</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E18:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E85</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <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>22</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E10:60_2_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E19:60_2_1_1_2_1_1_2_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font> &amp; 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <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>22</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E20:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E18:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>26</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#T8" target="_self">Th8</a></i>;<br/><a NAME="E22:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E15:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_5.html#E17:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E23:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><a NAME="E24:60_2_1_1_2_1_1_2_2_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font> <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>20</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>25</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E25:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:60_2_1_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_5.html#E23:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E26:60_2_1_1_2_1_1_2_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>26</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>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E8:60_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="graph_5.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E27:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E22:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E28:60_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:60_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E29:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E1:60_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_5.html#E4:60_2_1_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_5.html#E5:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_5.html#E18:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E30:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E31:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>26</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#E1:60_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_5.html#E27:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E89</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E33:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E34:60_2_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>27</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>2</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>2</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>10</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</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>11</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>26</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E15:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_5.html#E17:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_5.html#E33:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E90</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>28</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>2</sub></font><b> such that </b><br/><a NAME="E36:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</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>28</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font> )
 ;<br/><a NAME="E37:60_2_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E92</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E25:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_5.html#E36:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E38:60_2_1_1_2_1_1_2_2_1_2"/>
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</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>28</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E5:60_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_5.html#E36:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="graph_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E39:60_2_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>28</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>2</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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>9</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>9</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E36:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_5.html#E37:60_2_1_1_2_1_1_2_2_1_2"><i><font color="Green">E92</font></i></a></i>;<br/><b>hence </b><a NAME="E40:60_2_1_1_2_1_1_2_2_1_2"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>9</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></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:60"/><i><font color="Green">E53</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#E2:60"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_5.html#E3:60"><i><font color="Green">E52</font></i></a>);<br/></i>
<a NAME="E5:60"/>
<font color="Maroon">c<sub>5</sub></font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:60"><i><font color="Green">E50</font></i></a>, <a class="ref" href="graph_5.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:60"/><b>then </b>
<font color="Maroon">c<sub>5</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="E7:60"/><b>then </b>
 <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
 
<b>by </b><i><a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:60"/><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>5</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>5</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>hence </b><a NAME="E9:60"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>5</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#E1:60"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_5.html#E4:60"><i><font color="Green">E53</font></i></a></i>;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:60"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E12:60"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K5" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:60"><i><font color="Green">E50</font></i></a>, <a class="ref" href="graph_5.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E13:60"/>
 <a href="graph_5.html#K5" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</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#E11:60"><i><font color="Green">E54</font></i></a></i>;<br/>


</div>
