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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">W</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:228_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_1"><b>suppose </b></a><a NAME="E1:228_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:228_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:228_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E4:228_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>hence </b><a NAME="E5:228_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_2"><b>suppose </b></a><a NAME="E1:228_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:228_1_1_2"/><b>then </b>
not <font color="Maroon">W</font> is <a href="glib_001.html#V3" target="_self">trivial</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">lenW2</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:228_1_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">lenW2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a></i>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font>;<br/><b>set </b><font color="Maroon">vs1</font> = <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> ;<br/><a NAME="E5:228_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">lenW2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:228_1_1_2"/><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">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lenW2</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:228_1_1_2"/><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a>  &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a>  ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:228_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:228_1_1_2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:228_1_1_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E6:228_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a></i>;<br/><a NAME="E7:228_1_1_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E8:228_1_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E9:228_1_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:228_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div><a NAME="E10:228_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:228_1_1_2_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/><b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:228_1_1_2_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:228_1_1_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><i><font color="Green">E52</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> = 
<font color="Maroon">x1</font> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a></i>
<br/>.= 
<font color="Maroon">b1</font>
<b>by </b><i><a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><i><font color="Green">E53</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> = 
<font color="Maroon">x2</font> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><a NAME="E8:228_1_1_2_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</font>, <font color="Maroon">a2</font> = <font color="Maroon">a2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a></i>;<br/><div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:228_1_1_2_2_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> )
 ;<br/><a NAME="E2:228_1_1_2_2_3"/><b>then </b><i><font color="Green">E90</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1, <font color="Maroon">b</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E4:228_1_1_2_2_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:228_1_1_2_2_3"/><b>then </b><i><font color="Green">E91</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E6:228_1_1_2_2_3"/><i><font color="Green">E92</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, , <a class="txt" href="glib_001.html#E8:228_1_1_2_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E7:228_1_1_2_2_3"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E8:228_1_1_2_2_3"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E9:228_1_1_2_2_3"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenW2</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:228_1_1_2_2_3"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E11:228_1_1_2_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</font></i></a>, , , <a class="txt" href="glib_001.html#E59"><i><font color="Green">Lemma167</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:228_1_1_2_2_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="228_1_1_2_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_2_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:228_1_1_2_2_4_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_2_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:228_1_1_2_2_4_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:228_1_1_2_2_4_1_1_1"/><b>then </b>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:228_1_1_2_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, , <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma160</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="228_1_1_2_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:228_1_1_2_2_4_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:228_1_1_2_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, , <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma160</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:228_1_1_2_2_4"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:228_1_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, , <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma160</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E12:228_1_1_2"/><b>then </b>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E13:228_1_1_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/><a NAME="E14:228_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/><a NAME="E15:228_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D26">GLIB_000:def 26</a></i>;<br/><a NAME="E16:228_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E17:228_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">lenW2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E18:228_1_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenW2</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E19:228_1_1_2"/>
( not 1 is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenW2</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/><a NAME="E20:228_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenW2</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, </i>;<br/><a NAME="E21:228_1_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, <a class="ref" href="glib_001.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E22:228_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>hence </b><a NAME="E23:228_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:228"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K11" target="_self">.vertexSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>


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