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<div class="add">

<b>let </b><font color="Maroon">G</font> be  <a href="graph_1.html#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">vs</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">vs</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:47_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font>
 <b>and </b><br/><a NAME="E2:47_1"/><i><font color="Green">E37</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E3:47_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/><a NAME="E4:47_1"/><b>then </b><i><font color="Green">E46</font></i>: 
(  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:47_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E7:47_1"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i><a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:47_1"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/><a NAME="E9:47_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="graph_3.html#E2:47_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E10:47_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E11:47_1"/><b>then </b><i><font color="Green">E73</font></i>: 
0 <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</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:47_1"/>
<font color="Maroon">c</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="msscyc_1.html#D1">MSSCYC_1:def 1</a></i>;<br/><a NAME="E13:47_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">ic</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>;<br/><a NAME="E2:47_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:47_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:47_1_1_1"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</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</font></span>)</span></span>)</span> or <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</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</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E5:47_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, <a class="txt" href="graph_3.html#E4:47_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="graph_3.html#E6:47_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="graph_3.html#E7:47_1"><i><font color="Green">E48</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 ;<br/><div class="add"><a NAME="E2:47_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:47_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:47_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:47_1_1_2"/>
( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> or <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, , <a class="txt" href="graph_3.html#E6:47_1"><i><font color="Green">E47</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:47_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, <a class="txt" href="graph_3.html#E4:47_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="graph_3.html#E7:47_1"><i><font color="Green">E48</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E4:47"/><i><font color="Green">E81</font></i>: 
 <a href="graph_3.html#K7" target="_self">Degree</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E5:47"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_2"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1"><b>suppose </b></a><a NAME="E1:47_2_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K2" target="_self">Edges_In</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:47_2_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:47_2_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E7:47_2_1"/><i><font color="Green">E93</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:47_2_1"/><b>then </b><i><font color="Green">E94</font></i>: 
( ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) or ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, </i>;<br/><a NAME="E9:47_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</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">vs</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, , </i>;<br/><a NAME="E10:47_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:47_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2"><b>suppose </b></a><a NAME="E1:47_2_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_2_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_3.html#K3" target="_self">Edges_Out</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:47_2_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &amp; the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:47_2_2"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E7:47_2_2"/><i><font color="Green">E126</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:47_2_2"/><b>then </b><i><font color="Green">E127</font></i>: 
( ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) or ( <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, </i>;<br/><a NAME="E9:47_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</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">vs</font> )
 <b>by </b><i><a class="txt" href="graph_3.html#E2"><i><font color="Green">Lemma38</font></i></a>, , </i>;<br/><a NAME="E10:47_2_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:47_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="graph_3.html#T2" target="_self">Th2</a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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