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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>









<b>assume </b><a NAME="E1:231"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_001.html#K14" target="_self">.edges()</a>  &amp; <font color="Maroon">W1</font> is <a href="glib_001.html#V8" target="_self">Cycle-like</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">n</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="E3:231"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</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> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</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> 2</span>)</span> &amp; <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">WA</font> = <font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">WB</font> = <font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font>;<br/>
<a NAME="E4:231"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">v2</font>,<font color="Maroon">W1</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">W1</font></span>)</span>
 
<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#E15:231"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E5:231"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">v2</font>,<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 
;<br/>
<a NAME="E6:231"/><i><font color="Green">E43</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<a NAME="E7:231"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <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">W1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 
<b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E8:231"/><b>then </b>
<font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">v1</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#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="txt" href="glib_001.html#E15:231"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E9:231"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">v1</font>
 
;<br/>
<a NAME="E10:231"/>
<font color="Maroon">W1</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 
<b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, </i>;<br/>
<a NAME="E11:231"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a> ,<font color="Maroon">v1</font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, </i>;<br/>
<a NAME="E12:231"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">v2</font>,<font color="Maroon">v1</font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, </i>;<br/>
<a NAME="E13:231"/><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  )
 
<b>by </b><i><a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, </i>;<br/>
<a NAME="E14:231"/>
<font color="Maroon">W1</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 
<b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, </i>;<br/>
<a NAME="E15:231"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">W1</font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="231_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:231_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:231_1"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">maa1</font> = <font color="Maroon">m</font> <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>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E5:231_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">maa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E6:231_1"/>
<font color="Maroon">maa1</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> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:231_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">maa1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">maa1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W1</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#E58"><i><font color="Green">Lemma163</font></i></a>, <a class="txt" href="glib_001.html#E13:231"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E8:231_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">maa1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:231_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:231_1"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">maa1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E11:231_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <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/><b>hence </b><a NAME="E12:231_1"/>
contradiction
 <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#E56"><i><font color="Green">Lemma160</font></i></a>, , , <a class="txt" href="glib_001.html#E59"><i><font color="Green">Lemma167</font></i></a>, <a class="txt" href="glib_001.html#E17:231"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="231_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:231_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:231_2"/><i><font color="Green">E92</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i/>;<br/><a NAME="E4:231_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></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="E5:231_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</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="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</font></i></a>, </i>;<br/><a NAME="E6:231_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, </i>;<br/><a NAME="E7:231_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <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#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:231_2"/>
<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">W1</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:231_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma160</font></i></a>, <a class="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</font></i></a>, , , <a class="txt" href="glib_001.html#E17:231"><i><font color="Green">E91</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:231"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D30" target="_self">Def30</a></i>;<br/>
<a NAME="E19:231"/><b>then </b><i><font color="Green">E95</font></i>: 
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 
<b>by </b><i><a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:231"/><b>then </b><i><font color="Green">E96</font></i>: 
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 
<b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a></i>;<br/>
<a NAME="E21:231"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_001.html#K6" target="_self">.reverse()</a>  <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</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="glib_001.html#D12" target="_self">Def12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="231_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="231_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:231_3_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> ) )
 </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#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="231_3_1_1"><b>suppose </b></a><a NAME="E1:231_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:231_3_1_1"/>
 ex <font color="Olive">W2</font> being   <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">W2</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">W2</font> <a href="glib_001.html#K14" target="_self">.edges()</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="231_3_1_2"><b>suppose </b></a><a NAME="E1:231_3_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:231_3_1_2"/>
 ex <font color="Olive">W2</font> being   <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">W2</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">W2</font> <a href="glib_001.html#K14" target="_self">.edges()</a>  )
 <b>by </b><i><a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, <a class="txt" href="glib_001.html#E58"><i><font color="Green">Lemma163</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="E23:231"/>
 ex <font color="Olive">W2</font> being   <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">W2</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">W2</font> <a href="glib_001.html#K14" target="_self">.edges()</a>  )
 ;<br/>


</div>
