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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:170"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:170"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>set </b><font color="Maroon">CSN</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">V2N</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CSM</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">V2M</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">m</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:170"/><i><font color="Green">E38</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E4:170"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="170_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:170_1"/>
( <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="170_1_1"><b>suppose </b></a><a NAME="E1:170_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="170_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="170_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:170_1_1_1_1"/>
( <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <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="170_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:170_1_1_1_1_1"/>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:170_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="170_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:170_1_1_1_1_2"/>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:170_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E3:170_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:170_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="170_1_2"><b>suppose </b></a><a NAME="E1:170_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">CSK</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">V2K</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">k</font></span>)</span>;<br/><b>set </b><font color="Maroon">CK1</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">V21</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:170_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:170_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E4:170_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E5:170_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:170_1_2"/>
not <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
