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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W2</font>, <font color="Olive" title="b2">W1</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  st not <font color="Olive" title="b1">W2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  &amp; <font color="Olive" title="b1">W2</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">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/><font color="Olive" title="b1">W2</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">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">W2</font>, <font color="Maroon" title="c3">W1</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">W2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  &amp; <font color="Maroon" title="c2">W2</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">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  implies <font color="Maroon" title="c2">W2</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">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:220"/><span class="lab"><font color="Green" title="E143">A1</font></span>: 
not <font color="Maroon" title="c2">W2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">and </span><br/><a NAME="E2:220"/><span class="lab"><font color="Green" title="E144">A2</font></span>: 
<font color="Maroon" title="c2">W2</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">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">W2</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">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/>

<a NAME="E3:220"/><span class="lab"><font color="Green" title="E145">A3</font></span>: 
3 <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">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:220"><span class="lab"><font color="Green" title="E143">A1</font></span></a>, <a class="txt" href="glib_001.html#E91"><span class="lab"><font color="Green" title="E83">Lm54</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  implies <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:220_1"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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> <span class="kw"> such that </span><br/><a NAME="E3:220_1"/><span class="lab"><font color="Green" title="E146">A4</font></span>: 
<font color="Maroon" title="c5">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">W2</font>
 <span class="kw">and </span><br/><a NAME="E4:220_1"/><span class="lab"><font color="Green" title="E147">A5</font></span>: 
<font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E82"><span class="lab"><font color="Green" title="E74">Lm45</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="220_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:220_1_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W2</font> or <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:220_1_1_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/><div class="add"><a NAME="E2:220_1_1_1_1"/><span class="kw">then </span>
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:220"><span class="lab"><font color="Green" title="E145">A3</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">n5</font> = <font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as   <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><a NAME="E4:220_1_1_1_1"/><span class="lab"><font color="Green" title="E148">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:220_1_1_1_1"/>
<font color="Maroon" title="c6">n5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E6:220_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A7</font></span>: 
<font color="Maroon" title="c6">n5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:220_1"><span class="lab"><font color="Green" title="E146">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:220_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E150">A8</font></span>: 
<font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n5</font>,<font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" target="_self" title="GLIB_001:def.3">Def3</a></span>;<br/><a NAME="E8:220_1_1_1_1"/>
<font color="Maroon" title="c6">n5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:220_1_1_1_1"><span class="lab"><font color="Green" title="E149">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:220_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:220_1_1_1_1"><span class="lab"><font color="Green" title="E148">A6</font></span></a>, <a class="txt" href="glib_001.html#E83"><span class="lab"><font color="Green" title="E75">Lm46</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">m</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> <span class="kw"> such that </span><br/><a NAME="E11:220_1_1_1_1"/><span class="lab"><font color="Green" title="E151">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 <span class="kw">and </span><br/><a NAME="E12:220_1_1_1_1"/><span class="lab"><font color="Green" title="E152">A10</font></span>: 
<font color="Maroon" title="c7">m</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="c3">W1</font>
 <span class="kw">and </span><br/><a NAME="E13:220_1_1_1_1"/><span class="lab"><font color="Green" title="E153">A11</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:220"><span class="lab"><font color="Green" title="E144">A2</font></span></a>, <a class="txt" href="glib_001.html#E83"><span class="lab"><font color="Green" title="E75">Lm46</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">maa1</font> = <font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:220_1_1_1_1"><span class="lab"><font color="Green" title="E151">A9</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><a NAME="E15:220_1_1_1_1"/><span class="lab"><font color="Green" title="E154">A12</font></span>: 
<font color="Maroon" title="c8">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:220_1_1_1_1"><span class="lab"><font color="Green" title="E152">A10</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E16:220_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A13</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">maa1</font>,<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" target="_self" title="GLIB_001:def.3">Def3</a></span>;<br/><a NAME="E17:220_1_1_1_1"/><span class="lab"><font color="Green" title="E156">A14</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">maa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <font color="Maroon" title="c8">maa1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:220_1_1_1_1"><span class="lab"><font color="Green" title="E154">A12</font></span></a>, <a class="ref" href="glib_001.html#D8" target="_self" title="GLIB_001:def.8">Def8</a></span>;<br/><a NAME="E18:220_1_1_1_1"/><span class="lab"><font color="Green" title="E157">A15</font></span>: 
<font color="Maroon" title="c8">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c3">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:220_1_1_1_1"><span class="lab"><font color="Green" title="E154">A12</font></span></a>, <a class="ref" href="glib_001.html#T1" target="_self" title="GLIB_001:th.1">Th1</a></span>;<br/><a NAME="E19:220_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D8" target="_self" title="GLIB_001:def.8">Def8</a></span>;<br/><a NAME="E20:220_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <font color="Maroon" title="c8">maa1</font> or <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:220_1"><span class="lab"><font color="Green" title="E147">A5</font></span></a>, <a class="txt" href="#E7:220_1_1_1_1"><span class="lab"><font color="Green" title="E150">A8</font></span></a>, <a class="txt" href="#E13:220_1_1_1_1"><span class="lab"><font color="Green" title="E153">A11</font></span></a>, <a class="txt" href="#E16:220_1_1_1_1"><span class="lab"><font color="Green" title="E155">A13</font></span></a>, <a class="txt" href="#E17:220_1_1_1_1"><span class="lab"><font color="Green" title="E156">A14</font></span></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></span>;<br/><span class="kw">hence </span><a NAME="E21:220_1_1_1_1"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:220_1_1_1_1"><span class="lab"><font color="Green" title="E154">A12</font></span></a>, <a class="txt" href="#E18:220_1_1_1_1"><span class="lab"><font color="Green" title="E157">A15</font></span></a>, <a class="ref" href="glib_001.html#T90" target="_self" title="GLIB_001:th.90">Th90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="220_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:220_1_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/><div class="add"><a NAME="E2:220_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E148">A16</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:220_1"><span class="lab"><font color="Green" title="E146">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:220_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T101" target="_self" title="GLIB_001:th.101">Th101</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">m</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> <span class="kw"> such that </span><br/><a NAME="E5:220_1_1_1_2"/><span class="lab"><font color="Green" title="E149">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E6:220_1_1_1_2"/><span class="lab"><font color="Green" title="E150">A18</font></span>: 
<font color="Maroon" title="c6">m</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="c3">W1</font>
 <span class="kw">and </span><br/><a NAME="E7:220_1_1_1_2"/><span class="lab"><font color="Green" title="E151">A19</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:220"><span class="lab"><font color="Green" title="E144">A2</font></span></a>, <a class="txt" href="glib_001.html#E83"><span class="lab"><font color="Green" title="E75">Lm46</font></span></a></span>;<br/><a NAME="E8:220_1_1_1_2"/><span class="lab"><font color="Green" title="E152">A20</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:220_1"><span class="lab"><font color="Green" title="E147">A5</font></span></a>, <a class="txt" href="#E2:220_1_1_1_2"><span class="lab"><font color="Green" title="E148">A16</font></span></a>, <a class="txt" href="#E7:220_1_1_1_2"><span class="lab"><font color="Green" title="E151">A19</font></span></a>, <a class="ref" href="glib_001.html#D3" target="_self" title="GLIB_001:def.3">Def3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">maa1</font> = <font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:220_1_1_1_2"><span class="lab"><font color="Green" title="E149">A17</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><a NAME="E10:220_1_1_1_2"/><span class="lab"><font color="Green" title="E153">A21</font></span>: 
<font color="Maroon" title="c7">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:220_1_1_1_2"><span class="lab"><font color="Green" title="E150">A18</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E11:220_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A22</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">maa1</font>,<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" target="_self" title="GLIB_001:def.3">Def3</a></span>;<br/><a NAME="E12:220_1_1_1_2"/><span class="lab"><font color="Green" title="E155">A23</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">maa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <font color="Maroon" title="c7">maa1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:220_1_1_1_2"><span class="lab"><font color="Green" title="E153">A21</font></span></a>, <a class="ref" href="glib_001.html#D8" target="_self" title="GLIB_001:def.8">Def8</a></span>;<br/><a NAME="E13:220_1_1_1_2"/><span class="lab"><font color="Green" title="E156">A24</font></span>: 
<font color="Maroon" title="c7">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="c3">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:220_1_1_1_2"><span class="lab"><font color="Green" title="E153">A21</font></span></a>, <a class="ref" href="glib_001.html#T1" target="_self" title="GLIB_001:th.1">Th1</a></span>;<br/><a NAME="E14:220_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D8" target="_self" title="GLIB_001:def.8">Def8</a></span>;<br/><a NAME="E15:220_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <font color="Maroon" title="c7">maa1</font> or <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:220_1_1_1_2"><span class="lab"><font color="Green" title="E152">A20</font></span></a>, <a class="txt" href="#E11:220_1_1_1_2"><span class="lab"><font color="Green" title="E154">A22</font></span></a>, <a class="txt" href="#E12:220_1_1_1_2"><span class="lab"><font color="Green" title="E155">A23</font></span></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></span>;<br/><span class="kw">hence </span><a NAME="E16:220_1_1_1_2"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:220_1_1_1_2"><span class="lab"><font color="Green" title="E153">A21</font></span></a>, <a class="txt" href="#E13:220_1_1_1_2"><span class="lab"><font color="Green" title="E156">A24</font></span></a>, <a class="ref" href="glib_001.html#T90" target="_self" title="GLIB_001:th.90">Th90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:220_1"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:220"/>
<font color="Maroon" title="c2">W2</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">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
