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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">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">W</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font>;<br/>
<b>assume </b><a NAME="E1:96"/><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>
 ;<br/>

<a NAME="E2:96"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:96_1"/><i><font color="Green">E42</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#NR3" target="_self">&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> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> )
 ;<br/><a NAME="E2:96_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:96_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:96_1"/><b>then </b><i><font color="Green">E43</font></i>: 
2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:96_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 3
 <b>by </b><i>, , </i>;<br/><a NAME="E6:96_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:96_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:96"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:96_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&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> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:96_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:96_2"/><i><font color="Green">E46</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E4:96_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:96_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <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="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:96_2"/><b>then </b>
2 <a href="nat_1.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="E7:96_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E8:96_2"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 1 <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="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:96_2"/><b>then </b><i><font color="Green">E47</font></i>: 
3 <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="E10:96_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:96_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:96_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:96_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:96_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:96_2_1"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:96_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:96_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:96"/>
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i>, </i>;<br/>


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