<?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>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E34</font></i>: 
<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/>

<b>then reconsider </b><font color="Maroon">s</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>, <font color="Maroon">t</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="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</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/>
<b>reconsider </b><font color="Maroon">E</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><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 <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<a NAME="E5:12"/>
<span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="lang1.html#K3">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E7:12"/>
<font color="Maroon">c</font> is <a href="graph_1.html#V8">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a><br/>

<b>assume </b><a NAME="E1:12_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 ;<br/>

<b>hence </b><a NAME="E2:12_1"/>
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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">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">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
<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> is  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>
 ;<br/>


</div>
