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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">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">c<sub>1</sub></font> is <a href="msscyc_1.html#V2" target="_self">empty</a> or  not <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V2" target="_self">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">c<sub>1</sub></font> is <a href="msscyc_1.html#V2" target="_self">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="msscyc_1.html#V2" target="_self">empty</a> <a href="graph_1.html#NM1">Graph</a> ;<br/><a NAME="E3:37_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="msscyc_1.html#V3" target="_self">directed_cycle-less</a>
 ;<br/><b>hence </b><a NAME="E4:37_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V4" target="_self">well-founded</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V3" target="_self">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">c<sub>1</sub></font> is <a href="msscyc_1.html#V2" target="_self">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="msscyc_1.html#V2" target="_self">empty</a> <a href="graph_1.html#NM1">Graph</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E3:37_1_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V4" target="_self">well-founded</a>
 <b>and </b><br/><a NAME="E4:37_1_1_1_2"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="msscyc_1.html#V3" target="_self">directed_cycle-less</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:37_1_1_1_2"/><i><font color="Green">E22</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="msscyc_1.html#V1" target="_self">cyclic</a> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E4:37_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msscyc_1.html#D4" target="_self">Def4</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E7:37_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:37_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msscyc_1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E8:37_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:37_1_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:37_1_1_1_2"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>2</sub></font> st not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E3:37_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msscyc_1.html#D5" target="_self">Def5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:37_1_1_1_2"/><i><font color="Green">E24</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:37_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msscyc_1.html#T18" target="_self">Th18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msscyc_1.html#E14:37_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="msscyc_1.html#E14:37_1_1_1_2"><i><font color="Green">E24</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">vertex-seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</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">E22</font></i></a>, <a class="ref" href="msscyc_1.html#T17" target="_self">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">vertex-seq</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E6:37_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msscyc_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E19:37_1_1_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E12:37_1_1_1_2"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E20:37_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E14:37_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E21:37_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</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">E25</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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