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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Olive" title="b1">W1</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  holds <br/> ex <font color="Olive" title="b5">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 <br/>( <font color="Olive" title="b5">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> &amp;  not <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">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>  &amp; <font color="Maroon" title="c2">W1</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  holds <br/> ex <font color="Olive" title="b4">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 <br/>( <font color="Olive" title="b4">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> &amp;  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">e</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">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>  &amp; <font color="Maroon" title="c2">W1</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  implies  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp;  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:231"/><span class="lab"><font color="Green" title="E144">A1</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E2:231"/><span class="lab"><font color="Green" title="E145">A2</font></span>: 
<font color="Maroon" title="c3">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> 
 <span class="kw">and </span><br/><a NAME="E3:231"/><span class="lab"><font color="Green" title="E146">A3</font></span>: 
<font color="Maroon" title="c2">W1</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp;  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c8">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="E5:231"/><span class="lab"><font color="Green" title="E147">A4</font></span>: 
<font color="Maroon" title="c8">n</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="c2">W1</font>
 <span class="kw">and </span><br/><a NAME="E6:231"/><span class="lab"><font color="Green" title="E148">A5</font></span>: 
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">and </span><br/><a NAME="E7:231"/><span class="lab"><font color="Green" title="E149">A6</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:231"/><span class="lab"><font color="Green" title="E150">A7</font></span>: 
<font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:231"/><span class="lab"><font color="Green" title="E151">A8</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:231"><span class="lab"><font color="Green" title="E145">A2</font></span></a>, <a class="txt" href="glib_001.html#E84"><span class="lab"><font color="Green" title="E76">Lm47</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">WA</font> = <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">WB</font> = <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>);<br/>
<a NAME="E10:231"/><span class="lab"><font color="Green" title="E152">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:231"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="txt" href="glib_001.html#E53"><span class="lab"><font color="Green" title="E45">Lm16</font></span></a></span>;<br/>
<a NAME="E11:231"/><span class="lab"><font color="Green" title="E153">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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="c2">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="#E5:231"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E154">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="231_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:231_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span> <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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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="E3:231_1"/><span class="lab"><font color="Green" title="E155">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">and </span><br/><a NAME="E4:231_1"/><span class="lab"><font color="Green" title="E156">A13</font></span>: 
<font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:231_1"/><span class="lab"><font color="Green" title="E157">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E83"><span class="lab"><font color="Green" title="E75">Lm46</font></span></a></span>;<br/><a NAME="E6:231_1"/>
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:231_1"><span class="lab"><font color="Green" title="E155">A12</font></span></a>, <a class="txt" href="#E4:231_1"><span class="lab"><font color="Green" title="E156">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E7:231_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A15</font></span>: 
<font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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">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="#E7:231"><span class="lab"><font color="Green" title="E149">A6</font></span></a>, <a class="txt" href="#E11:231"><span class="lab"><font color="Green" title="E153">A10</font></span></a>, <a class="txt" href="#E5:231_1"><span class="lab"><font color="Green" title="E157">A14</font></span></a>, <a class="txt" href="glib_001.html#E60"><span class="lab"><font color="Green" title="E52">Lm23</font></span></a></span>;<br/><a NAME="E8:231_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:231"><span class="lab"><font color="Green" title="E153">A10</font></span></a>, <a class="txt" href="glib_001.html#E59"><span class="lab"><font color="Green" title="E51">Lm22</font></span></a></span>;<br/><a NAME="E9:231_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A16</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:231_1"><span class="lab"><font color="Green" title="E156">A13</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:231_1"/>
<font color="Maroon" title="c8">n</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">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:231"><span class="lab"><font color="Green" title="E153">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E11:231_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:231"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E3:231_1"><span class="lab"><font color="Green" title="E155">A12</font></span></a>, <a class="txt" href="#E7:231_1"><span class="lab"><font color="Green" title="E158">A15</font></span></a>, <a class="txt" href="#E9:231_1"><span class="lab"><font color="Green" title="E159">A16</font></span></a>, <a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E13:231"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E14:231"/><span class="kw">then </span>
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:231"><span class="lab"><font color="Green" title="E148">A5</font></span></a>, <a class="txt" href="#E11:231"><span class="lab"><font color="Green" title="E153">A10</font></span></a>, <a class="txt" href="glib_001.html#E53"><span class="lab"><font color="Green" title="E45">Lm16</font></span></a></span>;<br/>
<a NAME="E15:231"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A17</font></span>: 
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:231"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="ref" href="glib_001.html#D24" target="_self" title="GLIB_001:def.24">Def24</a></span>;<br/>
<a NAME="E16:231"/><span class="lab"><font color="Green" title="E156">A18</font></span>: 
<font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:231"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="txt" href="#E8:231"><span class="lab"><font color="Green" title="E150">A7</font></span></a>, <a class="txt" href="glib_001.html#E53"><span class="lab"><font color="Green" title="E45">Lm16</font></span></a></span>;<br/>
<a NAME="E17:231"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:231"><span class="lab"><font color="Green" title="E155">A17</font></span></a>, <a class="ref" href="glib_001.html#T31" target="_self" title="GLIB_001:th.31">Th32</a></span>;<br/>
<a NAME="E18:231"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T23" target="_self" title="GLIB_001:th.23">Th24</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E158">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="231_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:231_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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="E3:231_2"/><span class="lab"><font color="Green" title="E159">A21</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">and </span><br/><a NAME="E4:231_2"/><span class="lab"><font color="Green" title="E160">A22</font></span>: 
<font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:231_2"/><span class="lab"><font color="Green" title="E161">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><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="c12">maa1</font> = <font color="Maroon" title="c11">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="#E3:231_2"><span class="lab"><font color="Green" title="E159">A21</font></span></a>, <a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E7:231_2"/><span class="lab"><font color="Green" title="E162">A24</font></span>: 
<font color="Maroon" title="c12">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span></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="#E4:231_2"><span class="lab"><font color="Green" title="E160">A22</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E8:231_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">maa1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:231"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="txt" href="glib_001.html#E52"><span class="lab"><font color="Green" title="E44">Lm15</font></span></a></span>;<br/><a NAME="E9:231_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E163">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">maa1</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E10:231_2"/>
<font color="Maroon" title="c12">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font>
 ;<br/><a NAME="E11:231_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">A26</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">maa1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:231"><span class="lab"><font color="Green" title="E147">A4</font></span></a>, <a class="txt" href="#E5:231_2"><span class="lab"><font color="Green" title="E161">A23</font></span></a>, <a class="txt" href="#E7:231_2"><span class="lab"><font color="Green" title="E162">A24</font></span></a>, <a class="txt" href="glib_001.html#E52"><span class="lab"><font color="Green" title="E44">Lm15</font></span></a></span>;<br/><a NAME="E12:231_2"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:231_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">maa1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E14:231_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</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/><span class="kw">hence </span><a NAME="E15:231_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:231"><span class="lab"><font color="Green" title="E146">A3</font></span></a>, <a class="txt" href="#E7:231"><span class="lab"><font color="Green" title="E149">A6</font></span></a>, <a class="txt" href="#E11:231_2"><span class="lab"><font color="Green" title="E164">A26</font></span></a>, <a class="txt" href="#E9:231_2"><span class="lab"><font color="Green" title="E163">A25</font></span></a>, <a class="txt" href="#E13:231_2"><span class="lab"><font color="Green" title="E165">A27</font></span></a>, <a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E20:231"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:231"><span class="lab"><font color="Green" title="E155">A17</font></span></a>, <a class="ref" href="glib_001.html#D23" target="_self" title="GLIB_001:def.23">Def23</a></span>;<br/>
<a NAME="E21:231"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:231"><span class="lab"><font color="Green" title="E152">A9</font></span></a>, <a class="ref" href="glib_001.html#T102" target="_self" title="GLIB_001:th.102">Th103</a></span>;<br/>
<a NAME="E22:231"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A28</font></span>: 
 not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span></span>)</span> <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="#E19:231"><span class="lab"><font color="Green" title="E158">A20</font></span></a>, <a class="txt" href="#E12:231"><span class="lab"><font color="Green" title="E154">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E23:231"/><span class="kw">then </span><span class="lab"><font color="Green" title="E160">A29</font></span>: 
 not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W1</font></span>)</span>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">n</font>)</span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <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#T107" target="_self" title="GLIB_001:th.107">Th108</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="231_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="231_3_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="E1:231_3_1"/>
( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:231"><span class="lab"><font color="Green" title="E144">A1</font></span></a>, <a class="txt" href="#E9:231"><span class="lab"><font color="Green" title="E151">A8</font></span></a>, <a class="ref" href="glib_000.html#T15" title="GLIB_000:th.15">GLIB_000:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="231_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:231_3_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp;  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:231_3_1_1"/>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp;  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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="#E23:231"><span class="lab"><font color="Green" title="E160">A29</font></span></a>, <a class="txt" href="#E18:231"><span class="lab"><font color="Green" title="E157">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="231_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:231_3_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp;  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:231_3_1_2"/>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp;  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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="#E16:231"><span class="lab"><font color="Green" title="E156">A18</font></span></a>, <a class="txt" href="#E15:231"><span class="lab"><font color="Green" title="E155">A17</font></span></a>, <a class="txt" href="#E22:231"><span class="lab"><font color="Green" title="E159">A28</font></span></a>, <a class="ref" href="glib_001.html#T31" target="_self" title="GLIB_001:th.31">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E25:231"/>
 ex <font color="Olive" title="b1">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 <br/>( <font color="Olive" title="b1">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp;  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W2</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
