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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1"/>
( <font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> or  not <font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1"/>
<font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">G</font> = <font color="Maroon" title="c1">G</font> as  <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a> ;<br/><a NAME="E3:37_1_1_1_1"/>
<font color="Maroon" title="c2">G</font> is <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">directed_cycle-less</a>
 ;<br/><b>hence </b><a NAME="E4:37_1_1_1_1"/>
( <font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">well-founded</a> implies <font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">directed_cycle-less</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_2"/>
not <font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">G'</font> = <font color="Maroon" title="c1">G</font> as  non <a href="msscyc_1.html#V2" title="MSSCYC_1:attr.2">empty</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E3:37_1_1_1_2"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V4" title="MSSCYC_1:attr.4">well-founded</a>
 <b>and </b><br/><a NAME="E4:37_1_1_1_2"/><i><font color="Green" title="E21">A2</font></i>: 
not <font color="Maroon" title="c1">G</font> is <a href="msscyc_1.html#V3" title="MSSCYC_1:attr.3">directed_cycle-less</a>
 ;<br/><b>consider </b><font color="Maroon" title="c3">dc</font> being  <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="c2">G'</font><b> such that </b><br/><a NAME="E6:37_1_1_1_2"/><i><font color="Green" title="E22">A3</font></i>: 
( not <font color="Maroon" title="c3">dc</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c3">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#E4:37_1_1_1_2"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="msscyc_1.html#D4" target="_self" title="MSSCYC_1:def.4">Def4</a></i>;<br/><b>set </b><font color="Maroon" title="c4">p</font> =  <a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">dc</font>;<br/><a NAME="E7:37_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:37_1_1_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="msscyc_1.html#T10" target="_self" title="MSSCYC_1:th.10">Th10</a></i>;<br/><a NAME="E8:37_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:37_1_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">v</font> = <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 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="c1">G</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:37_1_1_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
for <font color="Olive" title="b1">c</font> being <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="c2">G'</font>  st not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:37_1_1_1_2"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="msscyc_1.html#D5" target="_self" title="MSSCYC_1:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">ch</font> being  <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="c2">G'</font><b> such that </b><br/><a NAME="E14:37_1_1_1_2"/><i><font color="Green" title="E24">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">ch</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">dc</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="c2">G'</font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:37_1_1_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="msscyc_1.html#T18" target="_self" title="MSSCYC_1:th.18">Th18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">ch</font> = <font color="Maroon" title="c7">ch</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="c2">G'</font> <b>by </b><i><a class="txt" href="msscyc_1.html#E14:37_1_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">cc</font> = <font color="Maroon" title="c8">ch</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">dc</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="c2">G'</font> <b>by </b><i><a class="txt" href="msscyc_1.html#E14:37_1_1_1_2"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/><a NAME="E17:37_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">dc</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:37_1_1_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="msscyc_1.html#T17" target="_self" title="MSSCYC_1:th.17">Th17</a></i>;<br/><a NAME="E18:37_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_2.html#K7" title="GRAPH_2:func.7">vertex-seq</a> <font color="Maroon" title="c9">cc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cc</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:37_1_1_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="msscyc_1.html#T16" target="_self" title="MSSCYC_1:th.16">Th16</a></i>;<br/><a NAME="E19:37_1_1_1_2"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E12:37_1_1_1_2"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/><a NAME="E20:37_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">dc</font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E14:37_1_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E21:37_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cc</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E22:37_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="msscyc_1.html#E19:37_1_1_1_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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