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

<b>set </b><font color="Maroon" title="c1">V</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c2">E</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">j</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">S</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c5">T</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">j</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">G</font> =  <a href="graph_1.html#G1" title="GRAPH_1:aggr.1">MultiGraphStruct</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font> #) as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a> <b>by </b><i><a class="ref" href="graph_1.html#D1" title="GRAPH_1:def.1">GRAPH_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">v</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G</font> ;<br/>
<a NAME="E5:38_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c4">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E6:38_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
not <font color="Maroon" title="c6">G</font> is <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">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" title="c8">dc</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> as  <a href="msscyc_1.html#NV1" title="MSSCYC_1:NV.1">directed</a>  <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c6">G</font> <b>by </b><i><a class="ref" href="msscyc_1.html#T5" target="_self" title="MSSCYC_1:th.5">Th5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">dc</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msscyc_1.html#D4" target="_self" title="MSSCYC_1:def.4">MSSCYC_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c8">dc</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">dc</font> is <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> )</span><br/>

<b>thus </b><a NAME="E2:38_1_1_1"/>
not <font color="Maroon" title="c8">dc</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">dc</font> is <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a></span><br/>

<a NAME="E3:38_1_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c8">dc</font>
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E5:38_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="msscyc_1.html#T4" target="_self" title="MSSCYC_1:th.4">Th4</a></i>;<br/>
<a NAME="E4:38_1_1_1"/>
( <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E5:38_1_1_1"/>
<font color="Maroon" title="c8">dc</font> is <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:38_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="msscyc_1.html#D2" target="_self" title="MSSCYC_1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c6">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c6">G</font> is <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">well-founded</a></span><br/>

<b>assume </b><a NAME="E7:38_1_1"/>
<font color="Maroon" title="c6">G</font> is <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">well-founded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c6">G</font> as  <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">well-founded</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a> ;<br/>
<a NAME="E9:38_1_1"/>
<font color="Maroon" title="c8">G</font> is <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">directed_cycle-less</a>
 
;<br/>
<b>hence </b><a NAME="E10:38_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:38_1_1"><i><font color="Green" title="E21">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
