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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/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><a NAME="E1:14"/><i><font color="Green">E15</font></i>: 
( the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) ) )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>6</sub></font> being  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:14"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:14"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="graphsp.html#E2:14"><i><font color="Green">E16</font></i></a>, <a class="ref" href="graph_5.html#D3">GRAPH_5:def 3</a></i>;<br/>
<a NAME="E4:14"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="graph_5.html#T8">GRAPH_5:8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> );<br/>
<a NAME="E5:14"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E6:14"/><i><font color="Green">E20</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="graphsp.html#E3:14"><i><font color="Green">E17</font></i></a>, <a class="txt" href="graphsp.html#E4:14"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:14"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="graphsp.html#E5:14"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graphsp.html#E6:14"><i><font color="Green">E20</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graphsp.html#E13"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<a NAME="E10:14"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="ref" href="graphsp.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:14"><i><font color="Green">E17</font></i></a>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/><b>hence </b><a NAME="E3:14_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graphsp.html#E10:14"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:14_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:14_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>;<br/><a NAME="E4:14_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_2_1"/><i><font color="Green">E27</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:14_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</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>6</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E2:14_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:14_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E4:14_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graphsp.html#E4:14_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graphsp.html#E1:14_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graphsp.html#E2:14_1_2_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:14_1_2"/><i><font color="Green">E27</font></i>: 
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graphsp.html#E13"><i><font color="Green">Lemma13</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graphsp.html#E13"><i><font color="Green">Lemma13</font></i></a></i>;<br/><a NAME="E8:14_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="graphsp.html#E3:14_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graphsp.html#E5:14_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graphsp.html#E6:14_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:14_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/><b>hence </b><a NAME="E10:14_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graphsp.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="graphsp.html#E8:14"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graphsp.html#E10:14"><i><font color="Green">E22</font></i></a>, <a class="txt" href="graphsp.html#E8:14_1_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
