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<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">vs1</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>let </b><font color="Maroon">vs2</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>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>let </b><font color="Maroon">c1</font> be    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c2</font> be    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">vs1</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c1</font>
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">vs2</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">vs1</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">vs1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> 1 )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:64"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font>
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">vs2</font>
 ;<br/>

<a NAME="E5:64"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">vs1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">vs2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>set </b><font color="Maroon">q</font> = <font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">vs1</font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">vs2</font>;<br/>
<a NAME="E7:64"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">vs1</font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">vs2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font></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">vs2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:64"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">vs1</font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">vs2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
;<br/>
<b>then </b><i><font color="Green">E42</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">vs1</font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">vs2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">vs1</font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">vs2</font> as    <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:64_2"/><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:64_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font></span>)</span>
 ;<br/><a NAME="E3:64_2"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</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">p</font> )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:64_2"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E5:64_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1"><b>suppose </b></a><a NAME="E1:64_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">v1</font> = <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">v2</font> = <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:64_2_1_1"/><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:64_2_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>,<font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:64_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:64_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</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">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:64_2_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</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">vs1</font> )
 <b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E7:64_2_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">vs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:64_2_1_1"/>
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E9:64_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>,<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2"><b>suppose </b></a><a NAME="E1:64_2_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:64_2_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E4:64_2_1_2"/><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:64_2_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font> &amp; <font color="Maroon">k</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">vs2</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E5:64_2_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>set </b><font color="Maroon">v1</font> = <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">v2</font> = <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E6:64_2_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:64_2_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>,<font color="Maroon">vs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:64_2_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:64_2_1_2"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:64_2_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:64_2_1_2"/><i><font color="Green">E64</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E12:64_2_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2_1_2_1_1"/>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</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="64_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:64_2_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:64_2_1_2_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs1</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:64_2_1_2"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E3:64_2_1_2_1_1_1"/><b>then </b>
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">vs1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:64_2_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="graph_2.html#E5:64_2_1_2"><i><font color="Green">E60</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:64_2_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:64_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>thus </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">p</font> <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">vs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E5:64_2_1_2"><i><font color="Green">E60</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E13:64_2_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs2</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:64_2_1_2"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">vs2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E5:64_2_1_2"><i><font color="Green">E60</font></i></a>, , , </i>
;<br/><b>hence </b><a NAME="E15:64_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="graph_1.html#R2">joins</a> <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>,<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E12:64_2_1_2"><i><font color="Green">E65</font></i></a>, , , , <a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:64"/>
<font color="Maroon">vs</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/>


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