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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W1</font> is    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Olive" title="b2">W2</font> holds <br/>( <font color="Olive" title="b1">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Olive" title="b1">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">W1</font> is    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c3">W2</font> implies ( <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  ) )</span><br/>





<b>assume </b><a NAME="E1:234"/><i><font color="Green" title="E146">A1</font></i>: 
<font color="Maroon" title="c2">W1</font> is    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c3">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">es</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span><b> such that </b><br/><a NAME="E3:234"/><i><font color="Green" title="E147">A2</font></i>: 
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c4">es</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D32" target="_self" title="GLIB_001:def.32">Def32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="234_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  implies <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )</span><br/><b>assume </b><a NAME="E1:234_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span><br/><b>then consider </b><font color="Maroon" title="c6">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:234_1"/><i><font color="Green" title="E148">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E83"><i><font color="Green" title="E75">Lm46</font></i></a></i>;<br/><a NAME="E4:234_1"/>
( <font color="Maroon" title="c6">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c4">es</font></span>)</span> &amp; <font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c4">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E3:234"><i><font color="Green" title="E147">A2</font></i></a>, <a class="txt" href="glib_001.html#E3:234_1"><i><font color="Green" title="E148">A3</font></i></a>, <a class="txt" href="glib_001.html#E77"><i><font color="Green" title="E69">Lm40</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:234_1"/><i><font color="Green" title="E149">A4</font></i>: 
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T4" target="_self" title="GLIB_001:th.4">Th4</a></i>;<br/><b>thus </b><a NAME="E7:234_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E3:234_1"><i><font color="Green" title="E148">A3</font></i></a>, <a class="txt" href="glib_001.html#E6:234_1"><i><font color="Green" title="E149">A4</font></i></a>, <a class="ref" href="glib_001.html#T105" target="_self" title="GLIB_001:th.105">Th105</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:234"/><i><font color="Green" title="E148">A5</font></i>: 
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="234_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="234_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:234_2_1"/>
( <font color="Maroon" title="c2">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> or  not <font color="Maroon" title="c2">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="234_2_1_1"><b>suppose </b></a><a NAME="E1:234_2_1_1"/><i><font color="Green" title="E149">A6</font></i>: 
<font color="Maroon" title="c2">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="234_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )</span><br/><b>assume </b><a NAME="E1:234_2_1_1_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/><b>then consider </b><font color="Maroon" title="c6">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:234_2_1_1_1"/><i><font color="Green" title="E150">A7</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font> &amp; <font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E82"><i><font color="Green" title="E74">Lm45</font></i></a></i>;<br/><a NAME="E4:234_2_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:234_2_1_1"><i><font color="Green" title="E149">A6</font></i></a>, <a class="txt" href="glib_001.html#E3:234_2_1_1_1"><i><font color="Green" title="E150">A7</font></i></a>, <a class="txt" href="glib_001.html#E92"><i><font color="Green" title="E84">Lm55</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:234_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E3:234_2_1_1_1"><i><font color="Green" title="E150">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:234_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E1:234"><i><font color="Green" title="E146">A1</font></i></a>, <a class="ref" href="glib_001.html#T162" target="_self" title="GLIB_001:th.162">Th162</a></i>;<br/><b>hence </b><a NAME="E7:234_2_1_1_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T89" target="_self" title="GLIB_001:th.89">Th89</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:234_2_1_1"/>
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="234_2_1_2"><b>suppose </b></a><a NAME="E1:234_2_1_2"/>
not <font color="Maroon" title="c2">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:234_2_1_2"/>
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E5:234"><i><font color="Green" title="E148">A5</font></i></a>, <a class="ref" href="glib_001.html#T135" target="_self" title="GLIB_001:th.135">Th135</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:234"/>
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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