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

<b>set </b><font color="Maroon">V</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">E</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E1:38_1_1"/><i><font color="Green">E34</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:38_1_1"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> 1, <font color="Maroon">T</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> 1 as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> =  <a href="graph_1.html#G1">MultiGraphStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">S</font>,<font color="Maroon">T</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/>
<b>reconsider </b><font color="Maroon">v</font> = 1 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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:38_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:38_1_1"/><i><font color="Green">E43</font></i>: 
not <font color="Maroon">G</font> is <a href="msscyc_1.html#V3" target="_self">directed_cycle-less</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">dc</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">dc</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msscyc_1.html#D4" target="_self">MSSCYC_1:def 4</a><br/>

<b>thus </b><a NAME="E2:38_1_1_1"/>
not <font color="Maroon">dc</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<a NAME="E3:38_1_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">v</font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">dc</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:38_1_1_1"/>
( <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">v</font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">v</font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">v</font></span><a href="lang1.html#K4">*&gt;</a></span> <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/>
<b>hence </b><a NAME="E5:38_1_1_1"/>
<font color="Maroon">dc</font> is <a href="msscyc_1.html#V1" target="_self">cyclic</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>assume </b><a NAME="E8:38_1_1"/>
<font color="Maroon">G</font> is <a href="msscyc_1.html#V4" target="_self">well-founded</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  <a href="msscyc_1.html#V4" target="_self">well-founded</a> <a href="graph_1.html#NM1">Graph</a> ;<br/>
<a NAME="E10:38_1_1"/>
<font color="Maroon">G</font> is <a href="msscyc_1.html#V3" target="_self">directed_cycle-less</a>
 
;<br/>
<b>hence </b><a NAME="E11:38_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
