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

<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/>

<b>set </b><font color="Maroon">E</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">S</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">T</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<b>thus </b><a NAME="E1:26"/>
( <font color="Maroon">G</font> is <a href="graph_1.html#V6">connected</a> implies for <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> holds <br/> ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G</font> is <a href="graph_1.html#V6">connected</a>
 ;<br/>

<b>let </b><font color="Maroon">v1</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E2:26_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 ;<br/>

<b>assume </b><a NAME="E3:26_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font><br/> for <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>  holds <br/>( <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> or  not <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  not <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">V</font> = the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>set </b><font color="Maroon">V1</font> = <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">V2</font> = <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) ) </span>}</span> ;<br/>
<a NAME="E5:26_1"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">V1</font> = <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E7:26_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">v</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">V</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_1"/>
<font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 ;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) ) )
 ;<br/><b>thus </b><a NAME="E4:26_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">V2</font> = <font color="Maroon">V</font> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> );<br/>
<b>consider </b><font color="Maroon">E1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:26_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E1</font> iff ( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">E2</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font>;<br/>
<a NAME="E12:26_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">E1</font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:26_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E1</font>
 ;<br/>

<b>hence </b><a NAME="E2:26_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E13:26_1"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:26_1"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:26_1"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K2">rng</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E16:26_1"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K2">rng</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<i><font color="Green">E77</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E1</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">E1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E18:26_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:26_1_4"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1_4"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> &amp; <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:26_1_4"/>
<span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E2:26_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:26_1_4"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="txt" href="graph_3.html#E2:26_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">S1</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E1</font>,<font color="Maroon">V1</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<i><font color="Green">E80</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E1</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">E1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">E1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E21:26_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:26_1_6"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1_6"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> &amp; <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:26_1_6"/>
<span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:26_1_6"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E2:26_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T1</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">E1</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E1</font>,<font color="Maroon">V1</font> <b>by </b><i><a class="txt" href="graph_3.html#E2:26_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<i><font color="Green">E82</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E24:26_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:26_1_8"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1_8"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E5:26_1_8"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:26_1_8"/><i><font color="Green">E91</font></i>: 
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E1</font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_8_1"/>
(  not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> or  not the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> or  not the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> )
 </span><b>by </b><i>, <a class="txt" href="graph_3.html#E3:26_1"><i><font color="Green">E36</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_8_1_1"><b>suppose </b></a><a NAME="E1:26_1_8_1_1"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_1_8_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_8_1_2"><b>suppose </b></a><a NAME="E1:26_1_8_1_2"/><i><font color="Green">E93</font></i>: 
not the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 ;<br/><div class="add"><a NAME="E2:26_1_8_1_2"/>
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E3:26_1_8_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 <b>by </b><i>, , <a class="txt" href="graph_3.html#E7:26_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_8_1_3"><b>suppose </b></a><a NAME="E1:26_1_8_1_3"/><i><font color="Green">E94</font></i>: 
not the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">Te</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:26_1_8_1_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:26_1_8_1_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/><a NAME="E5:26_1_8_1_3"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a></i>;<br/><b>assume </b><a NAME="E6:26_1_8_1_3"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 ;<br/><a NAME="E7:26_1_8_1_3"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">v'</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:26_1_8_1_3"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> ) ) )
 ;<br/><b>thus </b><a NAME="E10:26_1_8_1_3"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_8_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_8_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_8_1_3_1_1"/>
( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_8_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_8_1_3_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">ec</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="msscyc_1.html#T5">MSSCYC_1:5</a></i>;<br/><b>reconsider </b><font color="Maroon">vs</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">Te</font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><a NAME="E4:26_1_8_1_3_1_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">ec</font>
 <b>by </b><i>, , , , <a class="ref" href="msscyc_1.html#T4">MSSCYC_1:4</a></i>;<br/><a NAME="E5:26_1_8_1_3_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E6:26_1_8_1_3_1_1_1"/><b>then </b>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Te</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E7:26_1_8_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E7:26_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_8_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_8_1_3_1_1_2"/>
 ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c</font> being    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>, <font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:26_1_8_1_3_1_1_2"/><i><font color="Green">E128</font></i>: 
( not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> )
 ;<br/><b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , , , , </i>;<br/><a NAME="E5:26_1_8_1_3_1_1_2"/>
 ex <font color="Olive">vs'</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="graph_2.html#K3">^'</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Olive">vs'</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c'</font> &amp; <font color="Olive">vs'</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Olive">vs'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Te</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , , , , </i>;<br/><b>hence </b><a NAME="E6:26_1_8_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E7:26_1"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">S2</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font>,<font color="Maroon">V2</font> <b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<i><font color="Green">E131</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E27:26_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_10"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:26_1_10"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1_10"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/>
<a NAME="E5:26_1_10"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:26_1_10"/><i><font color="Green">E134</font></i>: 
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E1</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_10_1"/>
(  not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> or  not the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> or  not the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> )
 </span><b>by </b><i>, <a class="txt" href="graph_3.html#E7:26_1"><i><font color="Green">E37</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_10_1_1"><b>suppose </b></a><a NAME="E1:26_1_10_1_1"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_1_10_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_10_1_2"><b>suppose </b></a><a NAME="E1:26_1_10_1_2"/><i><font color="Green">E135</font></i>: 
not the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 ;<br/><div class="add"><a NAME="E2:26_1_10_1_2"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E3:26_1_10_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E3:26_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_10_1_3"><b>suppose </b></a><a NAME="E1:26_1_10_1_3"/><i><font color="Green">E136</font></i>: 
not the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">Se</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:26_1_10_1_3"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:26_1_10_1_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/><a NAME="E5:26_1_10_1_3"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E6:26_1_10_1_3"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 ;<br/><a NAME="E7:26_1_10_1_3"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E1:26_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">v'</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:26_1_10_1_3"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; ( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> ) ) )
 ;<br/><b>thus </b><a NAME="E10:26_1_10_1_3"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_10_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_10_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_10_1_3_1_1"/>
( <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or  ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_10_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_10_1_3_1_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">ec</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, , <a class="ref" href="msscyc_1.html#T5">MSSCYC_1:5</a></i>;<br/><b>reconsider </b><font color="Maroon">vs</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">Se</font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><a NAME="E4:26_1_10_1_3_1_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">ec</font>
 <b>by </b><i>, <a class="txt" href="graph_3.html#E3:26_1"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E5:26_1_10_1_3_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E6:26_1_10_1_3_1_1_1"/><b>then </b>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Se</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E7:26_1_10_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_10_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_10_1_3_1_1_2"/>
 ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c</font> being    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>, <font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:26_1_10_1_3_1_1_2"/><i><font color="Green">E143</font></i>: 
( not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> )
 ;<br/><b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, , <a class="txt" href="graph_3.html#E3:26_1"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E5:26_1_10_1_3_1_1_2"/>
 ex <font color="Olive">vs'</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">vs'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="graph_2.html#K3">^'</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>,<span class="p2">(<span class="default">the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Olive">vs'</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c'</font> &amp; <font color="Olive">vs'</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Olive">vs'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Se</font> )
 <b>by </b><i>, , <a class="txt" href="graph_3.html#E3:26_1"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E6:26_1_10_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T2</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font>,<font color="Maroon">V2</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G1</font> =  <a href="graph_1.html#G1">MultiGraphStruct</a>(# <font color="Maroon">V1</font>,<font color="Maroon">E1</font>,<font color="Maroon">S1</font>,<font color="Maroon">T1</font> #), <font color="Maroon">G2</font> =  <a href="graph_1.html#G1">MultiGraphStruct</a>(# <font color="Maroon">V2</font>,<span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font></span>)</span>,<font color="Maroon">S2</font>,<font color="Maroon">T2</font> #) as   <a href="graph_1.html#NM1">Graph</a> <b>by </b><i><a class="ref" href="graph_1.html#D1">GRAPH_1:def 1</a></i>;<br/>
<a NAME="E30:26_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">V1</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">V2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E31:26_1"/>
<font color="Maroon">G</font> <a href="graph_1.html#R1">is_sum_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_11"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:26_1_11"/><i><font color="Green">E148</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="graph_1.html#D3">GRAPH_1:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E1:26_1_11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font></span>)</span>
 ;<br/>

<a NAME="E2:26_1_11_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:26_1_11_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_11_1"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:26_1_11"/><i><font color="Green">E149</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E1:26_1_11_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font></span>)</span>
 ;<br/>

<a NAME="E2:26_1_11_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:26_1_11_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">E1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_11_2"/>
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>reconsider </b><font color="Maroon">MG</font> =  <a href="graph_1.html#G1">MultiGraphStruct</a>(# the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> #) as  <a href="graph_1.html#V1">strict</a> <a href="graph_1.html#NM1">Graph</a> <b>by </b><i><a class="ref" href="graph_1.html#D1">GRAPH_1:def 1</a></i>;<br/>
<a NAME="E4:26_1_11"/><i><font color="Green">E151</font></i>: 
the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">MG</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E5:26_1_11"/><i><font color="Green">E152</font></i>: 
the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">MG</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E6:26_1_11"/><i><font color="Green">E153</font></i>: 
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G1</font> holds <br/>( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">MG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">MG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E7:26_1_11"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G2</font> holds <br/>( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">MG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">MG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:26_1_11"/>
 <a href="graph_1.html#G1">MultiGraphStruct</a>(# the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="graph_1.html#K1">\/</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E3:26_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_3.html#E7:26_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, , , <a class="ref" href="graph_1.html#D2">GRAPH_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E32:26_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="graph_1.html#D7">GRAPH_1:def 7</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:26"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v2</font> holds <br/> ex <font color="Olive">c</font> being   <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> ex <font color="Olive">vs</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> st <br/>( not <font color="Olive">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Olive">c</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">v1</font> &amp; <font color="Olive">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v2</font> )
 ;<br/>

<b>thus </b><a NAME="E3:26"/>
<font color="Maroon">G</font> is <a href="graph_1.html#V6">connected</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">G1</font> being   <a href="graph_1.html#NM1">Graph</a>, <font color="Maroon">G2</font> being   <a href="graph_1.html#NM1">Graph</a><b> such that </b><a NAME="E1:26_2"/><i><font color="Green">E155</font></i>: 
the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font> <a href="subset_1.html#R1">misses</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font>
 <b>and </b><br/><a NAME="E2:26_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">G</font> <a href="graph_1.html#R1">is_sum_of</a> <font color="Maroon">G1</font>,<font color="Maroon">G2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_1.html#D7">GRAPH_1:def 7</a><br/>

<a NAME="E3:26_2"/><i><font color="Green">E157</font></i>: 
the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K3">/\</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">V1</font> = the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">V2</font> = the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font>;<br/>
<b>set </b><font color="Maroon">E1</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">S1</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">S2</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font>;<br/>
<b>set </b><font color="Maroon">T1</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">T2</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font>;<br/>
<a NAME="E4:26_2"/><i><font color="Green">E158</font></i>: 
( the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font> &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font> &amp;  <a href="graph_1.html#G1">MultiGraphStruct</a>(# the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="graph_1.html#K1">\/</a> <font color="Maroon">G2</font> )
 
<b>by </b><i>, <a class="ref" href="graph_1.html#D3">GRAPH_1:def 3</a></i>;<br/>
<a NAME="E5:26_2"/><b>then </b><i><font color="Green">E159</font></i>: 
( the <a href="graph_1.html#U1">Vertices</a> of <a href="graph_1.html#G1">MultiGraphStruct</a>(# the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>,the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font> &amp; the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G2</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G1</font> holds <br/>( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> ) ) &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G2</font> holds <br/>( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> ) ) )
 
<b>by </b><i><a class="ref" href="graph_1.html#D2">GRAPH_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">v1</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G1</font>;<br/>
<b>consider </b><font color="Maroon">v2</font> being   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G2</font>;<br/>
<b>reconsider </b><font color="Maroon">v1'</font> = <font color="Maroon">v1</font>, <font color="Maroon">v2'</font> = <font color="Maroon">v2</font> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:26_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>, <font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E11:26_2"/><i><font color="Green">E162</font></i>: 
( not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1'</font> &amp; <font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2'</font> )
 <b>by </b><i/>;<br/>
<a NAME="E12:26_2"/><i><font color="Green">E163</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G2</font> );<br/>
<a NAME="E13:26_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E14:26_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:26_2"/><b>then </b><i><font color="Green">E164</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:26_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E18:26_2"/><i><font color="Green">E166</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<a NAME="E19:26_2"/><i><font color="Green">E167</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:26_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E21:26_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E22:26_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E24:26_2"/><i><font color="Green">E168</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/>
<a NAME="E25:26_2"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:26_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E27:26_2"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon">e</font> = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>
<a NAME="E28:26_2"/>
<font color="Maroon">c</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="msscyc_1.html#D1">MSSCYC_1:def 1</a></i>;<br/>
<a NAME="E29:26_2"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E30:26_2"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E31:26_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E32:26_2"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1"/>
( <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G1</font> or <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G2</font> )
 </span><b>by </b><i>, , <a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:26_2_1_1"/><b>then </b><i><font color="Green">E175</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:26_2_1_1"/><i><font color="Green">E176</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>thus </b><a NAME="E4:26_2_1_1"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1_1_1_1"/>
( ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_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">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) or ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) )
 </span><b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1_1_1_1"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_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">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:26_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , , <a class="ref" href="graph_1.html#D2">GRAPH_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:26_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_1_1_1_2"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:26_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , , <a class="ref" href="graph_1.html#D2">GRAPH_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:26_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G2</font>
 ;<br/><div class="add"><a NAME="E2:26_2_1_2"/><b>then </b><i><font color="Green">E178</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E3:26_2_1_2"/><i><font color="Green">E179</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>thus </b><a NAME="E4:26_2_1_2"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_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="E1:26_2_1_2_1_1"/>
( ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_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">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) or ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) )
 </span><b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_2_1_1_1"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_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">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:26_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font>
 <b>by </b><i>, , , <a class="ref" href="graph_1.html#D2">GRAPH_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:26_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_2_1_1_2"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:26_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G2</font>
 <b>by </b><i>, , , <a class="ref" href="graph_1.html#D2">GRAPH_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:26_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="graph_3.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div>
