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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="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <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>;<br/>





<b>assume </b><a NAME="E1:88"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="graphsp.html#R5" target="_self">is_oriented_edge_seq_of</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<b>then </b><i><font color="Green">E73</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:88"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:88_2"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:88_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>5</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>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_1_1"><b>suppose </b></a><a NAME="E1:88_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:88_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>5</sub></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<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:88"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</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>4</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>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:88"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:88"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graphsp.html#E2:88_2_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_1_2"><b>suppose </b></a><a NAME="E1:88_2_1_2"/>
<font color="Maroon">c<sub>5</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>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:88_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:88_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:88_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</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>2</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:88_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graphsp.html#E1:88"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E5:88_2_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</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>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:88"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graphsp.html#E1:88_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:88"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graphsp.html#E1:88_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graphsp.html#E5:88_2_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graphsp.html#D19" target="_self">Def19</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:88"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E2:88"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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