<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="helly.html#NM1" title="HELLY:NM.1">_Tree</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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">P1</font>, <font color="Olive" title="b4">P2</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b4">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">P2</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>; <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">P1</font>, <font color="Olive" title="b2">P2</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Olive" title="b2">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> holds <br/><font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P2</font></span><br/><b>let </b><font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c5">P2</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c5">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font></span><br/>

<a NAME="E3:31"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c4">P1</font> <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">a</font>
 
<b>by </b><i><a class="txt" href="helly.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c5">P2</font> <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">a</font>
 
<b>by </b><i><a class="txt" href="helly.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<b>assume </b><a NAME="E5:31"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:31"/><i><font color="Green" title="E29">A6</font></i>: 
not <font color="Maroon" title="c4">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P1</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="c5">P2</font>
 
<b>by </b><i><a class="txt" href="helly.html#E5:31"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="finseq_2.html#T149" title="FINSEQ_2:th.149">FINSEQ_2:149</a></i>;<br/>
<a NAME="E3:31_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P1</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="c5">P2</font>
 
<b>by </b><i><a class="txt" href="helly.html#E1:31_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="finseq_1.html#T84" title="FINSEQ_1:th.84">FINSEQ_1:84</a></i>;<br/>
<a NAME="E4:31_1"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P1</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="c5">P2</font>
 
<b>by </b><i><a class="txt" href="helly.html#E2:31_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:31_1"/>
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="c4">P1</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/>
<a NAME="E6:31_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:31_1"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c4">P1</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="c4">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</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="c4">P1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="helly.html#E1:31_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<i><font color="Green" title="E34">A11</font></i>: <font color="Maroon" title="c4">P1</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="c4">P1</font></span>)</span> = 
<font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="helly.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/>
<font color="Maroon" title="c5">P2</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="c5">P2</font></span>)</span> = 
<font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="helly.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/>
<b>hence </b><a NAME="E10:31_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E4:31_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="helly.html#E7:31_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="helly.html#E8:31_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="helly.html#T25" target="_self" title="HELLY:th.25">Th25</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>
<a NAME="E7:31"/><i><font color="Green" title="E30">A12</font></i>: 
not <font color="Maroon" title="c5">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c5">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_2"/><b>then </b><i><font color="Green" title="E32">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P2</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="c4">P1</font>
 
<b>by </b><i><a class="txt" href="helly.html#E5:31"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="finseq_2.html#T149" title="FINSEQ_2:th.149">FINSEQ_2:149</a></i>;<br/>
<a NAME="E3:31_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P2</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="c4">P1</font>
 
<b>by </b><i><a class="txt" href="helly.html#E1:31_2"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="finseq_1.html#T84" title="FINSEQ_1:th.84">FINSEQ_1:84</a></i>;<br/>
<a NAME="E4:31_2"/><b>then </b><i><font color="Green" title="E33">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P2</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="c4">P1</font>
 
<b>by </b><i><a class="txt" href="helly.html#E2:31_2"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:31_2"/>
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="c5">P2</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/>
<a NAME="E6:31_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">P2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:31_2"/><b>then </b><i><font color="Green" title="E34">A16</font></i>: 
<font color="Maroon" title="c5">P2</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="c5">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</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="c5">P2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="helly.html#E1:31_2"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<i><font color="Green" title="E35">A17</font></i>: <font color="Maroon" title="c5">P2</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="c5">P2</font></span>)</span> = 
<font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="helly.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/>
<font color="Maroon" title="c4">P1</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="c4">P1</font></span>)</span> = 
<font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="helly.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/>
<b>hence </b><a NAME="E10:31_2"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E4:31_2"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="helly.html#E7:31_2"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="helly.html#E8:31_2"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="helly.html#T25" target="_self" title="HELLY:th.25">Th25</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>set </b><font color="Maroon" title="c6">di</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c5">P2</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">di</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c5">P2</font></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>  <b>by </b><i><a class="txt" href="helly.html#E3:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="helly.html#E4:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="helly.html#T21" target="_self" title="HELLY:th.21">Th21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">d</font> = <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( not $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; $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="c5">P2</font> &amp; <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  );<br/>
<a NAME="E9:31"/><i><font color="Green" title="E31">A18</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c9">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>]</span><br/>

<b>thus </b><a NAME="E1:31_3"/>
not <font color="Maroon" title="c9">k</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</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="c5">P2</font> &amp; <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )</span><br/>

<a NAME="E2:31_3"/><i><font color="Green" title="E32">A19</font></i>: 
<font color="Maroon" title="c7">di</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="c5">P2</font>
 
<b>by </b><i><a class="txt" href="helly.html#E3:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="helly.html#E4:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="helly.html#E7:31"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="helly.html#T22" target="_self" title="HELLY:th.22">Th22</a></i>;<br/>
<a NAME="E3:31_3"/>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E4:31_3"/>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="helly.html#E2:31_3"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="c9">k</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="c5">P2</font> &amp; <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )</span><br/>

<b>thus </b><a NAME="E5:31_3"/>
<font color="Maroon" title="c9">k</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="c5">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span><br/>

<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="helly.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<b>by </b><i><a class="txt" href="helly.html#E1:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/>
<b>hence </b><a NAME="E7:31_3"/>
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P1</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" title="GLIB_001:th.89">GLIB_001:89</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>consider </b><font color="Maroon" title="c9">ei</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:31"/><i><font color="Green" title="E32">A20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">ei</font>]
 <b>and </b><br/><a NAME="E12:31"/><i><font color="Green" title="E33">A21</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c9">ei</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="helly.html#E9:31"><i><font color="Green" title="E31">A18</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c10">ei</font> = <font color="Maroon" title="c9">ei</font> 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>  <b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>;<br/>
<b>set </b><font color="Maroon" title="c12">fi</font> = <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>;<br/>
<a NAME="E14:31"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span> <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="c4">P1</font>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></i>;<br/>
<a NAME="E15:31"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></i>;<br/>
<a NAME="E16:31"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_4"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:31_4"/>
contradiction
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</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>
<a NAME="E17:31"/><b>then </b><i><font color="Green" title="E36">A24</font></i>: 
not <font color="Maroon" title="c5">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></i>;<br/>
<a NAME="E18:31"/><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_5"/><i><font color="Green" title="E38">A26</font></i>: 
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_5"/><b>then </b><i><font color="Green" title="E39">A27</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="helly.html#T7" target="_self" title="HELLY:th.7">Th7</a></i>;<br/>
<a NAME="E3:31_5"/>
<font color="Maroon" title="c10">ei</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="helly.html#E1:31_5"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:31_5"/><b>then </b>
( <font color="Maroon" title="c5">P2</font> <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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> &amp; <font color="Maroon" title="c5">P2</font> <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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> )
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="helly.html#E15:31"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="helly.html#E2:31_5"><i><font color="Green" title="E39">A27</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/>
<a NAME="E5:31_5"/><b>then </b>
<font color="Maroon" title="c5">P2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/>
<a NAME="E6:31_5"/><b>then </b>
<font color="Maroon" title="c5">P2</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 
<b>by </b><i><a class="txt" href="helly.html#E17:31"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/>
<b>hence </b><a NAME="E7:31_5"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" title="GLIB_002:def.2">GLIB_002:def 2</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>set </b><font color="Maroon" title="c13">pde</font> = <font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font>;<br/>
<b>set </b><font color="Maroon" title="c14">pdf</font> = <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span>;<br/>
<a NAME="E19:31"/><i><font color="Green" title="E38">A28</font></i>: 
not <font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#T132" title="GLIB_001:th.132">GLIB_001:132</a></i>;<br/>
<a NAME="E20:31"/><b>then </b><i><font color="Green" title="E39">A29</font></i>: 
not <font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a></i>;<br/>
<a NAME="E21:31"/><i><font color="Green" title="E40">A30</font></i>: 
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font>
 
<b>by </b><i><a class="ref" href="helly.html#T7" target="_self" title="HELLY:th.7">Th7</a></i>;<br/>
<a NAME="E22:31"/><b>then </b><i><font color="Green" title="E41">A31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/>
<a NAME="E23:31"/><i><font color="Green" title="E42">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">rpdf</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> ;<br/>
<a NAME="E24:31"/><i><font color="Green" title="E43">A33</font></i>: 
not <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="glib_001.html#T132" title="GLIB_001:th.132">GLIB_001:132</a></i>;<br/>
<a NAME="E25:31"/><b>then </b>
not <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T130" title="GLIB_001:th.130">GLIB_001:130</a></i>;<br/>
<a NAME="E26:31"/><b>then </b><i><font color="Green" title="E44">A34</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a></i>;<br/>
<a NAME="E27:31"/>
not <font color="Maroon" title="c4">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="helly.html#E24:31"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="helly.html#T11" target="_self" title="HELLY:th.11">Th11</a></i>;<br/>
<a NAME="E28:31"/><b>then </b><i><font color="Green" title="E45">A35</font></i>: 
not <font color="Maroon" title="c4">P1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a></i>;<br/>
<a NAME="E29:31"/>
not <font color="Maroon" title="c5">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="helly.html#E19:31"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="helly.html#T11" target="_self" title="HELLY:th.11">Th11</a></i>;<br/>
<a NAME="E30:31"/><b>then </b><i><font color="Green" title="E46">A36</font></i>: 
not <font color="Maroon" title="c5">P2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 
<b>by </b><i><a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a></i>;<br/>
<a NAME="E31:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 
<b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E15:31"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/>
<a NAME="E32:31"/><b>then </b><i><font color="Green" title="E47">A37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 
<b>by </b><i><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></i>;<br/>
<a NAME="E33:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font>
 
<b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/>
<a NAME="E34:31"/><b>then </b><i><font color="Green" title="E48">A38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font>
 
<b>by </b><i><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></i>;<br/>
<a NAME="E35:31"/><i><font color="Green" title="E49">A39</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span> <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="c4">P1</font>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></i>;<br/>
<a NAME="E36:31"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T93" title="GLIB_001:th.93">GLIB_001:93</a></i>;<br/>
<a NAME="E37:31"/><b>then </b><i><font color="Green" title="E50">A40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <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="c4">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="helly.html#E35:31"><i><font color="Green" title="E49">A39</font></i></a>, <a class="ref" href="glib_001.html#T95" title="GLIB_001:th.95">GLIB_001:95</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">C</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span>;<br/>
<a NAME="E38:31"/><i><font color="Green" title="E51">A41</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_6_1"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c17">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_6_1"/><i><font color="Green" title="E52">A42</font></i>: 
<font color="Maroon" title="c17">g</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:31_6_1"/><i><font color="Green" title="E53">A43</font></i>: 
not <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:31_6_1"/>
<font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E3:31_6_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">gii</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="E7:31_6_1"/><i><font color="Green" title="E54">A44</font></i>: 
<font color="Maroon" title="c18">gii</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span>
 <b>and </b><br/><a NAME="E8:31_6_1"/><i><font color="Green" title="E55">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">gii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">gi</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:31_6_1"/><i><font color="Green" title="E56">A46</font></i>: 
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">gii</font>
 <b>and </b><br/><a NAME="E11:31_6_1"/><i><font color="Green" title="E57">A47</font></i>: 
<font color="Maroon" title="c19">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">gii</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>and </b><br/><a NAME="E12:31_6_1"/><i><font color="Green" title="E58">A48</font></i>: 
<font color="Maroon" title="c19">gi</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="c5">P2</font>
 <b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="helly.html#E7:31_6_1"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="helly.html#T12" target="_self" title="HELLY:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">gi</font> = <font color="Maroon" title="c19">gi</font> 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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E14:31_6_1"/><i><font color="Green" title="E59">A49</font></i>: 
<font color="Maroon" title="c18">gii</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="helly.html#E22:31"><i><font color="Green" title="E41">A31</font></i></a>, <a class="txt" href="helly.html#E4:31_6_1"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="helly.html#E8:31_6_1"><i><font color="Green" title="E55">A45</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E15:31_6_1"/>
<font color="Maroon" title="c18">gii</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/>
<a NAME="E16:31_6_1"/><b>then </b><i><font color="Green" title="E60">A50</font></i>: 
<font color="Maroon" title="c18">gii</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="helly.html#E14:31_6_1"><i><font color="Green" title="E59">A49</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E17:31_6_1"/><i><font color="Green" title="E61">A51</font></i>: 
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">gi</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_6_1_1"/>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c20">gi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_6_1_1"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">gii</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">gi</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="helly.html#E16:31_6_1"><i><font color="Green" title="E60">A50</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:31_6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E11:31_6_1"><i><font color="Green" title="E57">A47</font></i></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>
<a NAME="E18:31_6_1"/><i><font color="Green" title="E62">A52</font></i>: 
<font color="Maroon" title="c20">gi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">ei</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_6_1_2"/><i><font color="Green" title="E63">A53</font></i>: 
<font color="Maroon" title="c20">gi</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_6_1_2"/><i><font color="Green" title="E64">A54</font></i>: 
<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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_6_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_6_1_2_1"/>
( <font color="Maroon" title="c20">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ei</font> or <font color="Maroon" title="c20">gi</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">ei</font> )
 </span><b>by </b><i><a class="txt" href="helly.html#E1:31_6_1_2"><i><font color="Green" title="E63">A53</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_6_1_2_1_1"><b>suppose </b></a><a NAME="E1:31_6_1_2_1_1"/>
<font color="Maroon" title="c20">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">gii</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<b>by </b><i><a class="txt" href="helly.html#E11:31_6_1"><i><font color="Green" title="E57">A47</font></i></a>, <a class="txt" href="helly.html#E2:31_6_1_2"><i><font color="Green" title="E64">A54</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>
;<br/><b>hence </b><a NAME="E3:31_6_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E4:31_6_1"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="helly.html#E8:31_6_1"><i><font color="Green" title="E55">A45</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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="31_6_1_2_1_2"><b>suppose </b></a><a NAME="E1:31_6_1_2_1_2"/>
<font color="Maroon" title="c20">gi</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:31_6_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c20">gi</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:31_6_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E7:31_6_1"><i><font color="Green" title="E54">A44</font></i></a>, <a class="txt" href="helly.html#E11:31_6_1"><i><font color="Green" title="E57">A47</font></i></a>, <a class="txt" href="helly.html#E2:31_6_1_2"><i><font color="Green" title="E64">A54</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</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>
<a NAME="E19:31_6_1"/>
<font color="Maroon" title="c17">g</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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E3:31_6_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E20:31_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E12:31"><i><font color="Green" title="E33">A21</font></i></a>, <a class="txt" href="helly.html#E37:31"><i><font color="Green" title="E50">A40</font></i></a>, <a class="txt" href="helly.html#E8:31_6_1"><i><font color="Green" title="E55">A45</font></i></a>, <a class="txt" href="helly.html#E10:31_6_1"><i><font color="Green" title="E56">A46</font></i></a>, <a class="txt" href="helly.html#E12:31_6_1"><i><font color="Green" title="E58">A48</font></i></a>, <a class="txt" href="helly.html#E17:31_6_1"><i><font color="Green" title="E61">A51</font></i></a>, <a class="txt" href="helly.html#E18:31_6_1"><i><font color="Green" title="E62">A52</font></i></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>thus </b><a NAME="E2:31_6"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c17">x</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:31_6_2"/><i><font color="Green" title="E52">A55</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_6_2_1"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> )
 </span><b>by </b><i><a class="txt" href="helly.html#E1:31_6_2"><i><font color="Green" title="E52">A55</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_6_2_1_1"><b>suppose </b></a><a NAME="E1:31_6_2_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/><div class="add"><a NAME="E2:31_6_2_1_1"/><b>then </b>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c17">x</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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="helly.html#E23:31"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="helly.html#E32:31"><i><font color="Green" title="E47">A37</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><b>hence </b><a NAME="E3:31_6_2_1_1"/>
<font color="Maroon" title="c17">x</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0: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="31_6_2_1_2"><b>suppose </b></a><a NAME="E1:31_6_2_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span><br/><div class="add"><a NAME="E2:31_6_2_1_2"/><b>then </b>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c17">x</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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="helly.html#E22:31"><i><font color="Green" title="E41">A31</font></i></a>, <a class="txt" href="helly.html#E34:31"><i><font color="Green" title="E48">A38</font></i></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></i>;<br/><b>hence </b><a NAME="E3:31_6_2_1_2"/>
<font color="Maroon" title="c17">x</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0: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><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E39:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></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>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_7"/><i><font color="Green" title="E52">A56</font></i>: 
<font color="Maroon" title="c17">x</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></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>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:31_7"/><i><font color="Green" title="E53">A57</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E3:31_7"><i><font color="Green" title="E52">A56</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:31_7"/><i><font color="Green" title="E54">A58</font></i>: 
<font color="Maroon" title="c17">x</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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></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> 
 
<b>by </b><i><a class="txt" href="helly.html#E3:31_7"><i><font color="Green" title="E52">A56</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:31_7"/><i><font color="Green" title="E55">A59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></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> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T108" title="GLIB_001:th.108">GLIB_001:108</a></i>;<br/>
<a NAME="E7:31_7"/><i><font color="Green" title="E56">A60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T93" title="GLIB_001:th.93">GLIB_001:93</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">v1</font>, <font color="Maroon" title="c19">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c20">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="E9:31_7"/><i><font color="Green" title="E57">A61</font></i>: 
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span>
 <b>and </b><br/><a NAME="E10:31_7"/><i><font color="Green" title="E58">A62</font></i>: 
<font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font>
 <b>and </b><br/><a NAME="E11:31_7"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</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="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E12:31_7"/><i><font color="Green" title="E59">A63</font></i>: 
<font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</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="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E13:31_7"/><i><font color="Green" title="E60">A64</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">v1</font>,<font color="Maroon" title="c19">v2</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="helly.html#E4:31_7"><i><font color="Green" title="E53">A57</font></i></a>, <a class="ref" href="glib_001.html#T104" title="GLIB_001:th.104">GLIB_001:104</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">u1</font>, <font color="Maroon" title="c22">u2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c23">m</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="E15:31_7"/><i><font color="Green" title="E61">A65</font></i>: 
<font color="Maroon" title="c23">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E16:31_7"/><i><font color="Green" title="E62">A66</font></i>: 
<font color="Maroon" title="c21">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font>
 <b>and </b><br/><a NAME="E17:31_7"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E18:31_7"/><i><font color="Green" title="E63">A67</font></i>: 
<font color="Maroon" title="c22">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E19:31_7"/><i><font color="Green" title="E64">A68</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c21">u1</font>,<font color="Maroon" title="c22">u2</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="helly.html#E5:31_7"><i><font color="Green" title="E54">A58</font></i></a>, <a class="txt" href="helly.html#E6:31_7"><i><font color="Green" title="E55">A59</font></i></a>, <a class="ref" href="glib_001.html#T104" title="GLIB_001:th.104">GLIB_001:104</a></i>;<br/>
<a NAME="E20:31_7"/><i><font color="Green" title="E65">A69</font></i>: 
<font color="Maroon" title="c20">n</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="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_7_1"/>
( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">v2</font> or <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_1_1"><b>suppose </b></a><a NAME="E1:31_7_1_1"/><i><font color="Green" title="E66">A70</font></i>: 
<font color="Maroon" title="c18">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:31_7_1_1"/>
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E9:31_7"><i><font color="Green" title="E57">A61</font></i></a>, <a class="txt" href="helly.html#E20:31_7"><i><font color="Green" title="E65">A69</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:31_7_1_1"/><b>then </b><i><font color="Green" title="E67">A71</font></i>: 
<font color="Maroon" title="c18">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E10:31_7"><i><font color="Green" title="E58">A62</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E4:31_7_1_1"/><i><font color="Green" title="E68">A72</font></i>: 
<font color="Maroon" title="c19">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E9:31_7"><i><font color="Green" title="E57">A61</font></i></a>, <a class="txt" href="helly.html#E12:31_7"><i><font color="Green" title="E59">A63</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E5:31_7_1_1"/>
<font color="Maroon" title="c20">n</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="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:31_7_1_1"/><b>then </b><i><font color="Green" title="E69">A73</font></i>: 
<font color="Maroon" title="c20">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E9:31_7"><i><font color="Green" title="E57">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:31_7_1_1"/>
<font color="Maroon" title="c23">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="c23">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E8:31_7_1_1"/><b>then </b><i><font color="Green" title="E70">A74</font></i>: 
<font color="Maroon" title="c23">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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E15:31_7"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:31_7_1_1"/><b>then </b><i><font color="Green" title="E71">A75</font></i>: 
<font color="Maroon" title="c21">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E16:31_7"><i><font color="Green" title="E62">A66</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E10:31_7_1_1"/><i><font color="Green" title="E72">A76</font></i>: 
<font color="Maroon" title="c22">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E15:31_7"><i><font color="Green" title="E61">A65</font></i></a>, <a class="txt" href="helly.html#E18:31_7"><i><font color="Green" title="E63">A67</font></i></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E11:31_7_1_1"/><i><font color="Green" title="E73">A77</font></i>: 
( ( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u1</font> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">u2</font> ) or ( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">u2</font> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u1</font> ) )
 <b>by </b><i><a class="txt" href="helly.html#E13:31_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="helly.html#E19:31_7"><i><font color="Green" title="E64">A68</font></i></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></i>;<br/><a NAME="E12:31_7_1_1"/><i><font color="Green" title="E74">A78</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">v1</font>,<font color="Maroon" title="c19">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E3:31_7_1_1"><i><font color="Green" title="E67">A71</font></i></a>, <a class="txt" href="helly.html#E4:31_7_1_1"><i><font color="Green" title="E68">A72</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E13:31_7_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c21">u1</font>,<font color="Maroon" title="c22">u2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="helly.html#E7:31_7"><i><font color="Green" title="E56">A60</font></i></a>, <a class="txt" href="helly.html#E9:31_7_1_1"><i><font color="Green" title="E71">A75</font></i></a>, <a class="txt" href="helly.html#E10:31_7_1_1"><i><font color="Green" title="E72">A76</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/><a NAME="E14:31_7_1_1"/><b>then </b><i><font color="Green" title="E75">A79</font></i>: 
( ( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> or <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> ) &amp; ( <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> or <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> ) )
 <b>by </b><i><a class="txt" href="helly.html#E38:31"><i><font color="Green" title="E51">A41</font></i></a>, <a class="txt" href="helly.html#E11:31_7_1_1"><i><font color="Green" title="E73">A77</font></i></a>, <a class="txt" href="helly.html#E12:31_7_1_1"><i><font color="Green" title="E74">A78</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a>, <a class="ref" href="zfmisc_1.html#T28" title="ZFMISC_1:th.28">ZFMISC_1:28</a></i>;<br/><a NAME="E15:31_7_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">im</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E17:31_7_1_1"/><i><font color="Green" title="E76">A80</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">im</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">im</font> = <font color="Maroon" title="c24">im</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E19:31_7_1_1"/><i><font color="Green" title="E77">A81</font></i>: 
<font color="Maroon" title="c25">im</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E8:31_7_1_1"><i><font color="Green" title="E70">A74</font></i></a>, <a class="txt" href="helly.html#E17:31_7_1_1"><i><font color="Green" title="E76">A80</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E20:31_7_1_1"/><b>then </b><i><font color="Green" title="E78">A82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">im</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="helly.html#E17:31_7_1_1"><i><font color="Green" title="E76">A80</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E21:31_7_1_1"/><i><font color="Green" title="E79">A83</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">m</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
 ;<br/><a NAME="E22:31_7_1_1"/><b>then </b><i><font color="Green" title="E80">A84</font></i>: 
<font color="Maroon" title="c23">m</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E15:31_7"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E23:31_7_1_1"/><b>then </b><i><font color="Green" title="E81">A85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">m</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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="helly.html#E21:31_7_1_1"><i><font color="Green" title="E79">A83</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E24:31_7_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">ni</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E26:31_7_1_1"/><i><font color="Green" title="E82">A86</font></i>: 
<font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">ni</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">ni</font> = <font color="Maroon" title="c26">ni</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E28:31_7_1_1"/><i><font color="Green" title="E83">A87</font></i>: 
<font color="Maroon" title="c27">ni</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E6:31_7_1_1"><i><font color="Green" title="E69">A73</font></i></a>, <a class="txt" href="helly.html#E26:31_7_1_1"><i><font color="Green" title="E82">A86</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E29:31_7_1_1"/><b>then </b><i><font color="Green" title="E84">A88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">ni</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="helly.html#E26:31_7_1_1"><i><font color="Green" title="E82">A86</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E30:31_7_1_1"/><i><font color="Green" title="E85">A89</font></i>: 
<font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">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
 ;<br/><a NAME="E31:31_7_1_1"/><b>then </b><i><font color="Green" title="E86">A90</font></i>: 
<font color="Maroon" title="c20">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E9:31_7"><i><font color="Green" title="E57">A61</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E32:31_7_1_1"/><b>then </b><i><font color="Green" title="E87">A91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</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="c20">n</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="helly.html#E30:31_7_1_1"><i><font color="Green" title="E85">A89</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E33:31_7_1_1"/><i><font color="Green" title="E88">A92</font></i>: 
<font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_7_1_1_1_1"/>
( ( <font color="Maroon" title="c21">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> &amp; <font color="Maroon" title="c22">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> ) or ( <font color="Maroon" title="c21">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c22">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> ) )
 </span><b>by </b><i><a class="txt" href="helly.html#E13:31_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="helly.html#E19:31_7"><i><font color="Green" title="E64">A68</font></i></a>, <a class="txt" href="helly.html#E1:31_7_1_1"><i><font color="Green" title="E66">A70</font></i></a>, <a class="txt" href="helly.html#E14:31_7_1_1"><i><font color="Green" title="E75">A79</font></i></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_7_1_1_1_1_1"/><i><font color="Green" title="E89">A93</font></i>: 
( <font color="Maroon" title="c21">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> &amp; <font color="Maroon" title="c22">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span><br/><div class="add"><a NAME="E2:31_7_1_1_1_1_1"/>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:31_7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E90">A94</font></i>: 
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:31_7_1_1_1_1_1"/>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">di</font>
 <b>by </b><i><a class="txt" href="helly.html#E15:31_7"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:31_7_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">m</font></span>)</span> <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 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E6:31_7_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">m</font></span>)</span> <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> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:31_7_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c4">P1</font>
 <b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:31_7_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">P1</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c4">P1</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> )
 <b>by </b><i><a class="txt" href="helly.html#E18:31_7"><i><font color="Green" title="E63">A67</font></i></a>, <a class="txt" href="helly.html#E23:31_7_1_1"><i><font color="Green" title="E81">A85</font></i></a>, <a class="txt" href="helly.html#E1:31_7_1_1_1_1_1"><i><font color="Green" title="E89">A93</font></i></a>, <a class="txt" href="helly.html#E3:31_7_1_1_1_1_1"><i><font color="Green" title="E90">A94</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E9:31_7_1_1_1_1_1"/>
<font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 <b>by </b><i><a class="txt" href="helly.html#E28:31"><i><font color="Green" title="E45">A35</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</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="31_7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_7_1_1_1_1_2"/><i><font color="Green" title="E89">A95</font></i>: 
( <font color="Maroon" title="c21">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c22">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span><br/><div class="add"><a NAME="E2:31_7_1_1_1_1_2"/>
<font color="Maroon" title="c25">im</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_7_1_1_1_1_2_1"/><i><font color="Green" title="E90">A96</font></i>: 
<font color="Maroon" title="c25">im</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_7_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">im</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="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">di</font>
 
<b>by </b><i><a class="txt" href="helly.html#E19:31_7_1_1"><i><font color="Green" title="E77">A81</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:31_7_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">im</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/>
<a NAME="E4:31_7_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">im</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:31_7_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E91">A97</font></i>: 
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">im</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="c4">P1</font>
 
<b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:31_7_1_1_1_1_2_1"/>
<font color="Maroon" title="c23">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">im</font>
 
<b>by </b><i><a class="txt" href="helly.html#E17:31_7_1_1"><i><font color="Green" title="E76">A80</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c28">im</font> = <font color="Maroon" title="c25">im</font> as  <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>  ;<br/>
<a NAME="E8:31_7_1_1_1_1_2_1"/>
<font color="Maroon" title="c28">im</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E1:31_7_1_1_1_1_2_1"><i><font color="Green" title="E90">A96</font></i></a></i>;<br/>
<a NAME="E9:31_7_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">di</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="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">im</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:31_7_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">P1</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c4">P1</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> )
 
<b>by </b><i><a class="txt" href="helly.html#E16:31_7"><i><font color="Green" title="E62">A66</font></i></a>, <a class="txt" href="helly.html#E20:31_7_1_1"><i><font color="Green" title="E78">A82</font></i></a>, <a class="txt" href="helly.html#E1:31_7_1_1_1_1_2"><i><font color="Green" title="E89">A95</font></i></a>, <a class="txt" href="helly.html#E5:31_7_1_1_1_1_2_1"><i><font color="Green" title="E91">A97</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/>
<b>hence </b><a NAME="E11:31_7_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E28:31"><i><font color="Green" title="E45">A35</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</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="E3:31_7_1_1_1_1_2"/>
<font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 <b>by </b><i><a class="txt" href="helly.html#E14:31"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="helly.html#E18:31"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="helly.html#E18:31_7"><i><font color="Green" title="E63">A67</font></i></a>, <a class="txt" href="helly.html#E17:31_7_1_1"><i><font color="Green" title="E76">A80</font></i></a>, <a class="txt" href="helly.html#E22:31_7_1_1"><i><font color="Green" title="E80">A84</font></i></a>, <a class="txt" href="helly.html#E1:31_7_1_1_1_1_2"><i><font color="Green" title="E89">A95</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</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><a NAME="E34:31_7_1_1"/>
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_7_1_1_2_1"/>
( ( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> ) or ( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> ) )
 </span><b>by </b><i><a class="txt" href="helly.html#E1:31_7_1_1"><i><font color="Green" title="E66">A70</font></i></a>, <a class="txt" href="helly.html#E14:31_7_1_1"><i><font color="Green" title="E75">A79</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_7_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:31_7_1_1_2_1_1"/><i><font color="Green" title="E89">A98</font></i>: 
( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span><br/><div class="add"><a NAME="E2:31_7_1_1_2_1_1"/>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:31_7_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E90">A99</font></i>: 
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:31_7_1_1_2_1_1"/>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">di</font>
 <b>by </b><i><a class="txt" href="helly.html#E9:31_7"><i><font color="Green" title="E57">A61</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:31_7_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">n</font></span>)</span> <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 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E6:31_7_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">n</font></span>)</span> <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> <font color="Maroon" title="c10">ei</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:31_7_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c5">P2</font>
 <b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:31_7_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c5">P2</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c5">P2</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> )
 <b>by </b><i><a class="txt" href="helly.html#E21:31"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="helly.html#E12:31_7"><i><font color="Green" title="E59">A63</font></i></a>, <a class="txt" href="helly.html#E32:31_7_1_1"><i><font color="Green" title="E87">A91</font></i></a>, <a class="txt" href="helly.html#E1:31_7_1_1_2_1_1"><i><font color="Green" title="E89">A98</font></i></a>, <a class="txt" href="helly.html#E3:31_7_1_1_2_1_1"><i><font color="Green" title="E90">A99</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E9:31_7_1_1_2_1_1"/>
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 <b>by </b><i><a class="txt" href="helly.html#E30:31"><i><font color="Green" title="E46">A36</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</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="31_7_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:31_7_1_1_2_1_2"/><i><font color="Green" title="E89">A100</font></i>: 
( <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span><br/><div class="add"><a NAME="E2:31_7_1_1_2_1_2"/>
<font color="Maroon" title="c27">ni</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_7_1_1_2_1_2_1"/><i><font color="Green" title="E90">A101</font></i>: 
<font color="Maroon" title="c27">ni</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:31_7_1_1_2_1_2_1"/>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">ni</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="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">di</font>
 
<b>by </b><i><a class="txt" href="helly.html#E28:31_7_1_1"><i><font color="Green" title="E83">A87</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:31_7_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">ni</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/>
<a NAME="E4:31_7_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">ni</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">ei</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:31_7_1_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E91">A102</font></i>: 
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">ni</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="c5">P2</font>
 
<b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:31_7_1_1_2_1_2_1"/>
<font color="Maroon" title="c20">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">ni</font>
 
<b>by </b><i><a class="txt" href="helly.html#E26:31_7_1_1"><i><font color="Green" title="E82">A86</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c28">ni</font> = <font color="Maroon" title="c27">ni</font> as  <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>  ;<br/>
<a NAME="E8:31_7_1_1_2_1_2_1"/>
<font color="Maroon" title="c28">ni</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="helly.html#E1:31_7_1_1_2_1_2_1"><i><font color="Green" title="E90">A101</font></i></a></i>;<br/>
<a NAME="E9:31_7_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">di</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="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ni</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:31_7_1_1_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">P2</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> &amp; <font color="Maroon" title="c5">P2</font> <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="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">di</font> )
 
<b>by </b><i><a class="txt" href="helly.html#E21:31"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="helly.html#E10:31_7"><i><font color="Green" title="E58">A62</font></i></a>, <a class="txt" href="helly.html#E29:31_7_1_1"><i><font color="Green" title="E84">A88</font></i></a>, <a class="txt" href="helly.html#E1:31_7_1_1_2_1_2"><i><font color="Green" title="E89">A100</font></i></a>, <a class="txt" href="helly.html#E5:31_7_1_1_2_1_2_1"><i><font color="Green" title="E91">A102</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/>
<b>hence </b><a NAME="E11:31_7_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E30:31"><i><font color="Green" title="E46">A36</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</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="E3:31_7_1_1_2_1_2"/>
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">di</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="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font>
 <b>by </b><i><a class="txt" href="helly.html#E11:31"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="helly.html#E12:31_7"><i><font color="Green" title="E59">A63</font></i></a>, <a class="txt" href="helly.html#E26:31_7_1_1"><i><font color="Green" title="E82">A86</font></i></a>, <a class="txt" href="helly.html#E31:31_7_1_1"><i><font color="Green" title="E86">A90</font></i></a>, <a class="txt" href="helly.html#E1:31_7_1_1_2_1_2"><i><font color="Green" title="E89">A100</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</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="E35:31_7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E3:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="helly.html#E4:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="helly.html#E6:31"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="helly.html#E7:31"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="helly.html#E33:31_7_1_1"><i><font color="Green" title="E88">A92</font></i></a>, <a class="ref" href="helly.html#T23" target="_self" title="HELLY:th.23">Th23</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="31_7_1_2"><b>suppose </b></a><a NAME="E1:31_7_1_2"/>
<font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:31_7_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</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="c18">v1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></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="c18">v1</font> )
 <b>by </b><i><a class="txt" href="helly.html#E9:31_7"><i><font color="Green" title="E57">A61</font></i></a>, <a class="txt" href="helly.html#E10:31_7"><i><font color="Green" title="E58">A62</font></i></a>, <a class="txt" href="helly.html#E12:31_7"><i><font color="Green" title="E59">A63</font></i></a>, <a class="txt" href="helly.html#E20:31_7"><i><font color="Green" title="E65">A69</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E3:31_7_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="helly.html#E20:31"><i><font color="Green" title="E39">A29</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</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>
<a NAME="E40:31"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<font color="Maroon" title="c10">ei</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c7">di</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ei</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 
<b>by </b><i><a class="txt" href="helly.html#E20:31"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="helly.html#E22:31"><i><font color="Green" title="E41">A31</font></i></a>, <a class="txt" href="helly.html#E23:31"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="helly.html#E26:31"><i><font color="Green" title="E44">A34</font></i></a>, <a class="txt" href="helly.html#E32:31"><i><font color="Green" title="E47">A37</font></i></a>, <a class="txt" href="helly.html#E34:31"><i><font color="Green" title="E48">A38</font></i></a>, <a class="txt" href="helly.html#E38:31"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="helly.html#T19" target="_self" title="HELLY:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E41:31"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" title="GLIB_002:def.2">GLIB_002:def 2</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>
