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<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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#NM3" title="GRAPH_1:NM.3">Graph</a> ;<br/>
<span class="kw">reconsider </span><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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">G</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">G</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">G</font> is  <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a> </span><br/>

<a NAME="E5:35_1_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<a NAME="E6:35_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c6">G</font> is  <a href="msscyc_1.html#NV4" title="MSSCYC_1:NV.4">with_directed_cycle</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c8">dc</font> = <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default">1</span><a href="pre_poly.html#K3" title="PRE_POLY: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> <span class="kw">by</span> <span class="lab"><a class="ref" href="msscyc_1.html#T5" target="_self" onmouseover="rs('msscyc_1/T5')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">dc</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msscyc_1.html#D3" target="_self" onmouseover="rs('msscyc_1/D3')" onmouseout="rh()">MSSCYC_1:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E2:35_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:35_1_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
( <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="pre_poly.html#K4" title="PRE_POLY: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="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E4:35_1_1_1"/>
( <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="pre_poly.html#K4" title="PRE_POLY: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> &amp; <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font></span><a href="pre_poly.html#K4" title="PRE_POLY: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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="msscyc_1.html#T4" target="_self" onmouseover="rs('msscyc_1/T4')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:35_1_1_1"/>
<font color="Maroon" title="c8">dc</font> is  <a href="msscyc_1.html#V1" title="MSSCYC_1:attr.1">cyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E7:35_1_1"/>
<font color="Maroon" title="c6">G</font> is  <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c6">G</font> as   <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">well-founded</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a> ;<br/>
<a NAME="E9:35_1_1"/>
<font color="Maroon" title="c8">G</font> is  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">directed_cycle-less</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E10:35_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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