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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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/>
<a NAME="E3:32"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
( <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> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T22" target="_self" title="HELLY:th.22">Th22</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>(  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/>
<span class="kw">assume </span><a NAME="E5:32"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:32"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 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="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:32_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T63" title="FINSEQ_1:th.63">FINSEQ_1:63</a></span>;<br/>
<a NAME="E3:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A7</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="finseq_2.html#T129" title="FINSEQ_2:th.129">FINSEQ_2:129</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E4:32_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="c5">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E5:32_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E32">A9</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:32_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="helly.html#T26" target="_self" title="HELLY:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:32"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
 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="32_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c8">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E1:32_2"/>
 not <font color="Maroon" title="c8">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">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="c8">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="E2:32_2"/>
( <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> &amp; <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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="helly.html#T23" target="_self" title="HELLY:th.23">Th23</a>, <a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:32_2"/>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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="c8">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/>

<span class="kw">thus </span><a NAME="E4:32_2"/>
<font color="Maroon" title="c8">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c8">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="c8">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
<br/>.= 
<font color="Maroon" title="c4">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:32_2"/>
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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 class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">ei</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:32"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">ei</font>]
 <span class="kw">and </span><br/><a NAME="E10:32"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
 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="c8">ei</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E7:32"><span class="lab"><font color="Green" title="E29">A10</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c9">ei</font> = <font color="Maroon" title="c8">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">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="c9">ei</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">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="c9">ei</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">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="c9">ei</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">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="c9">ei</font></span>)</span></span>)</span>);<br/>
<a NAME="E12:32"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> ;<br/>
<a NAME="E13:32"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<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="c9">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T92" title="GLIB_001:th.92">GLIB_001:92</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">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="c9">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">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/>
<a NAME="E14:32"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T7" target="_self" title="HELLY:th.7">Th7</a></span>;<br/>
<a NAME="E15:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/>
<a NAME="E16:32"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">ei</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></span>;<br/>
<a NAME="E17:32"/>
 <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="32_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_3"/>
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:32_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">di</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:32_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A18</font></span>: 
 not <font color="Maroon" title="c5">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/>
<a NAME="E19:32"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<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="c9">ei</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_4"/>
<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="c9">ei</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:32_4"/><span class="kw">then </span>
( <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="c9">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="c9">ei</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">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="c9">ei</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="helly.html#T7" target="_self" title="HELLY:th.7">Th7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:32_4"/><span class="kw">then </span>
( <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="c9">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="c9">ei</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E16:32"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A20</font></span>: 
 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="c9">ei</font></span>)</span></span>)</span>) is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="glib_001.html#T131" title="GLIB_001:th.131">GLIB_001:131</a></span>;<br/>
<a NAME="E21:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A21</font></span>: 
 not <font color="Maroon" title="c4">P1</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
;<br/>
<a NAME="E22:32"/>
<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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#D19" title="GLIB_001:def.19">GLIB_001:def 19</a></span>;<br/>
<a NAME="E23:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A22</font></span>: 
<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="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E13:32"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="glib_001.html#T94" title="GLIB_001:th.94">GLIB_001:94</a></span>;<br/>
<a NAME="E24:32"/><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<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="c9">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="c9">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="c9">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="32_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_5"/>
 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="c9">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="c9">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="c9">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:32_5"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<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="c9">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="c9">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 class="kw">and </span><br/><a NAME="E4:32_5"/><span class="lab"><font color="Green" title="E44">A25</font></span>: 
 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="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E5:32_5"/>
<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="c9">ei</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_5"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E7:32_5"/><span class="lab"><font color="Green" title="E45">A26</font></span>: 
<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="c9">ei</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:32_5"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<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="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:32_5"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
<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="c9">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="kw">and </span><br/><a NAME="E11:32_5"/><span class="lab"><font color="Green" title="E48">A29</font></span>: 
<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
 <span class="kw">and </span><br/><a NAME="E12:32_5"/><span class="lab"><font color="Green" title="E49">A30</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E7:32_5"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="ref" href="helly.html#T12" target="_self" title="HELLY:th.12">Th12</a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E14:32_5"/><span class="lab"><font color="Green" title="E50">A31</font></span>: 
<font color="Maroon" title="c18">gii</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E15:32_5"/><span class="lab"><font color="Green" title="E51">A32</font></span>: 
<font color="Maroon" title="c20">gi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">ei</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_5_1"/><span class="lab"><font color="Green" title="E52">A33</font></span>: 
<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="c9">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="c9">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:32_5_1"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
<font color="Maroon" title="c20">gi</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_5_1_1"/>
( <font color="Maroon" title="c20">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c9">ei</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_5_1"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_5_1_1_1"/>
<font color="Maroon" title="c20">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><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="c9">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="c9">ei</font>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_5"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="txt" href="#E1:32_5_1"><span class="lab"><font color="Green" title="E52">A33</font></span></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">ei</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:32_5_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_5"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="txt" href="#E8:32_5"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_5_1_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="c9">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:32_5_1_1_2"/><span class="kw">then </span>
<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="c9">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_5_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_5"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="txt" href="#E11:32_5"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="txt" href="#E1:32_5_1"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E16:32_5"/>
<font color="Maroon" title="c18">gii</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E4:32_5"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="txt" href="#E8:32_5"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E17:32_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A35</font></span>: 
<font color="Maroon" title="c18">gii</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_5"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E18:32_5"/><span class="lab"><font color="Green" title="E53">A36</font></span>: 
<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="32_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_5_2"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:32_5_2"/><span class="kw">then </span>
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_5"><span class="lab"><font color="Green" title="E52">A35</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:32_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_5"><span class="lab"><font color="Green" title="E48">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:32_5"/>
<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="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_5"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:32_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E23:32"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="txt" href="#E8:32_5"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E10:32_5"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E12:32_5"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="txt" href="#E18:32_5"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="txt" href="#E15:32_5"><span class="lab"><font color="Green" title="E51">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:32"/><span class="lab"><font color="Green" title="E43">A37</font></span>: 
<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="c9">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="c9">ei</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/>
<a NAME="E26:32"/>
<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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/>
<a NAME="E27:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A38</font></span>: 
<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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T22" title="GLIB_001:th.22">GLIB_001:22</a></span>;<br/>
<a NAME="E28:32"/>
<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="c9">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="c9">ei</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E16:32"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/>
<a NAME="E29:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A39</font></span>: 
<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="c9">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="c9">ei</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T22" title="GLIB_001:th.22">GLIB_001:22</a></span>;<br/>
<a NAME="E30:32"/>
<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="c9">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="c9">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="c9">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c9">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="c9">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="c9">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/>

<span class="kw">assume </span><a NAME="E1:32_6"/><span class="lab"><font color="Green" title="E46">A40</font></span>: 
<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="c9">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c9">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="c9">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="32_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_6_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="c9">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_6"><span class="lab"><font color="Green" title="E46">A40</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_6_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="c9">ei</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="c9">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="c9">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="E2:32_6_1_1"/><span class="kw">then </span>
( <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="c9">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="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="txt" href="#E29:32"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_6_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="c9">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="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_6_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c9">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="c9">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="E2:32_6_1_2"/><span class="kw">then </span>
( <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="c9">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="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E27:32"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_6_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="c9">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="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E31:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A41</font></span>: 
<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="c9">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="c9">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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:32"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E32:32"/><span class="lab"><font color="Green" title="E47">A42</font></span>: 
 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="c9">ei</font>) is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#T131" title="GLIB_001:th.131">GLIB_001:131</a></span>;<br/>
<a NAME="E33:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A43</font></span>: 
 not <font color="Maroon" title="c5">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
;<br/>
<a NAME="E34:32"/><span class="lab"><font color="Green" title="E49">A44</font></span>: 
 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="32_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_7"/><span class="lab"><font color="Green" title="E50">A45</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:32_7"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T63" title="FINSEQ_1:th.63">FINSEQ_1:63</a></span>;<br/>
<a NAME="E3:32_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A46</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E1:32_7"><span class="lab"><font color="Green" title="E50">A45</font></span></a>, <a class="ref" href="finseq_2.html#T129" title="FINSEQ_2:th.129">FINSEQ_2:129</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E4:32_7"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E5:32_7"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:32_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A47</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_7"><span class="lab"><font color="Green" title="E50">A45</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A48</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:32_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_7"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="txt" href="#E6:32_7"><span class="lab"><font color="Green" title="E52">A47</font></span></a>, <a class="txt" href="#E7:32_7"><span class="lab"><font color="Green" title="E53">A48</font></span></a>, <a class="ref" href="helly.html#T26" target="_self" title="HELLY:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:32"/><span class="lab"><font color="Green" title="E50">A49</font></span>: 
<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="c9">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="c9">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="32_8"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_8"/><span class="lab"><font color="Green" title="E51">A50</font></span>: 
<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="c9">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="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T92" title="GLIB_001:th.92">GLIB_001:92</a></span>;<br/>
<a NAME="E2:32_8"/><span class="lab"><font color="Green" title="E52">A51</font></span>: 
<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="c9">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="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T107" title="GLIB_001:th.107">GLIB_001:107</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:32_8"/>
<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="c9">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="c9">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> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:32_8"/><span class="lab"><font color="Green" title="E53">A52</font></span>: 
<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="c9">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="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E6:32_8"/>
<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="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_8"><span class="lab"><font color="Green" title="E53">A52</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">u1</font>, <font color="Maroon" title="c19">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="c20">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> <span class="kw"> such that </span><br/><a NAME="E8:32_8"/><span class="lab"><font color="Green" title="E54">A53</font></span>: 
<font color="Maroon" title="c20">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:32_8"/><span class="lab"><font color="Green" title="E55">A54</font></span>: 
<font color="Maroon" title="c18">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font>
 <span class="kw">and </span><br/><a NAME="E10:32_8"/>
<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="c9">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="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:32_8"/><span class="lab"><font color="Green" title="E56">A55</font></span>: 
<font color="Maroon" title="c19">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="c9">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="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:32_8"/><span class="lab"><font color="Green" title="E57">A56</font></span>: 
<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">u1</font>,<font color="Maroon" title="c19">u2</font>,<font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_8"><span class="lab"><font color="Green" title="E52">A51</font></span></a>, <a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/>
<a NAME="E13:32_8"/>
<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="c9">ei</font>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_8"><span class="lab"><font color="Green" title="E53">A52</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">v1</font>, <font color="Maroon" title="c22">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="c23">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E15:32_8"/><span class="lab"><font color="Green" title="E58">A57</font></span>: 
<font color="Maroon" title="c23">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="c9">ei</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:32_8"/><span class="lab"><font color="Green" title="E59">A58</font></span>: 
<font color="Maroon" title="c21">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="c9">ei</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">and </span><br/><a NAME="E17:32_8"/>
<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="c9">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="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:32_8"/><span class="lab"><font color="Green" title="E60">A59</font></span>: 
<font color="Maroon" title="c22">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="c9">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="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:32_8"/><span class="lab"><font color="Green" title="E61">A60</font></span>: 
<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">v1</font>,<font color="Maroon" title="c22">v2</font>,<font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/>
<a NAME="E20:32_8"/><span class="lab"><font color="Green" title="E62">A61</font></span>: 
<font color="Maroon" title="c23">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="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_8_1"/>
( <font color="Maroon" title="c21">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">v2</font> or <font color="Maroon" title="c21">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_8_1_1"/><span class="lab"><font color="Green" title="E63">A62</font></span>: 
<font color="Maroon" title="c21">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:32_8_1_1"/><span class="lab"><font color="Green" title="E64">A63</font></span>: 
<font color="Maroon" title="c23">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="c23">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="E3:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A64</font></span>: 
<font color="Maroon" title="c23">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="c9">ei</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_8"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A65</font></span>: 
<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="c9">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="c23">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="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E2:32_8_1_1"><span class="lab"><font color="Green" title="E64">A63</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">ni</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:32_8_1_1"/><span class="lab"><font color="Green" title="E67">A66</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">ni</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">ni</font> = <font color="Maroon" title="c24">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E8:32_8_1_1"/><span class="lab"><font color="Green" title="E68">A67</font></span>: 
<font color="Maroon" title="c19">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_8"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="txt" href="#E11:32_8"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><a NAME="E9:32_8_1_1"/>
<font color="Maroon" title="c20">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="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A68</font></span>: 
<font color="Maroon" title="c20">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_8"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:32_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_8"><span class="lab"><font color="Green" title="E55">A54</font></span></a>, <a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><a NAME="E12:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A69</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">u1</font>,<font color="Maroon" title="c19">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="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_8"><span class="lab"><font color="Green" title="E51">A50</font></span></a>, <a class="txt" href="#E8:32_8_1_1"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E13:32_8_1_1"/><span class="lab"><font color="Green" title="E71">A70</font></span>: 
<font color="Maroon" title="c20">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="c20">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="E14:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A71</font></span>: 
<font color="Maroon" title="c20">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_8"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A72</font></span>: 
<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="c9">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="c20">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="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E13:32_8_1_1"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><a NAME="E16:32_8_1_1"/>
<font color="Maroon" title="c23">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="c9">ei</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_8"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="txt" href="#E20:32_8"><span class="lab"><font color="Green" title="E62">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A73</font></span>: 
<font color="Maroon" title="c21">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="c9">ei</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:32_8"><span class="lab"><font color="Green" title="E59">A58</font></span></a>, <a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><a NAME="E18:32_8_1_1"/>
<font color="Maroon" title="c22">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="c9">ei</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_8"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="txt" href="#E18:32_8"><span class="lab"><font color="Green" title="E60">A59</font></span></a>, <a class="ref" href="glib_001.html#T87" title="GLIB_001:th.87">GLIB_001:87</a></span>;<br/><a NAME="E19:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A74</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c21">v1</font>,<font color="Maroon" title="c22">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="c9">ei</font>)</span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_8_1_1"><span class="lab"><font color="Green" title="E74">A73</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E20:32_8_1_1"/><span class="lab"><font color="Green" title="E76">A75</font></span>: 
( ( <font color="Maroon" title="c21">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">u1</font> &amp; <font color="Maroon" title="c22">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u2</font> ) or ( <font color="Maroon" title="c21">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u2</font> &amp; <font color="Maroon" title="c22">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">u1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:32_8"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="txt" href="#E12:32_8"><span class="lab"><font color="Green" title="E57">A56</font></span></a>, <a class="ref" href="glib_000.html#T15" title="GLIB_000:th.15">GLIB_000:15</a></span>;<br/><a NAME="E21:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A76</font></span>: 
( <font color="Maroon" title="c21">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="c9">ei</font> or <font color="Maroon" title="c21">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:32"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="txt" href="#E19:32_8_1_1"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="txt" href="#E12:32_8_1_1"><span class="lab"><font color="Green" title="E70">A69</font></span></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#T22" title="ZFMISC_1:th.22">ZFMISC_1:22</a></span>;<br/><a NAME="E22:32_8_1_1"/>
<font color="Maroon" title="c23">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="c23">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E23:32_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">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="c9">ei</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_8"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E24:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A77</font></span>: 
<font color="Maroon" title="c25">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="c9">ei</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_8_1_1"><span class="lab"><font color="Green" title="E67">A66</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E25:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A78</font></span>: 
<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="c9">ei</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">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="c25">ni</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E6:32_8_1_1"><span class="lab"><font color="Green" title="E67">A66</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><a NAME="E26:32_8_1_1"/><span class="lab"><font color="Green" title="E80">A79</font></span>: 
<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="c9">ei</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_8_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_8_1_1_1_1"/>
( ( <font color="Maroon" title="c21">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="c9">ei</font> &amp; <font color="Maroon" title="c22">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="c21">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="c22">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="c9">ei</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:32"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="txt" href="#E1:32_8_1_1"><span class="lab"><font color="Green" title="E63">A62</font></span></a>, <a class="txt" href="#E20:32_8_1_1"><span class="lab"><font color="Green" title="E76">A75</font></span></a>, <a class="txt" href="#E19:32_8_1_1"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="txt" href="#E12:32_8_1_1"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="txt" href="#E21:32_8_1_1"><span class="lab"><font color="Green" title="E77">A76</font></span></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#T22" title="ZFMISC_1:th.22">ZFMISC_1:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_8_1_1_1_1_1"/><span class="lab"><font color="Green" title="E81">A80</font></span>: 
( <font color="Maroon" title="c21">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="c9">ei</font> &amp; <font color="Maroon" title="c22">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c9">ei</font></span><br/><div class="add"><a NAME="E2:32_8_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">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="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_8"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:32_8_1_1_1_1_1"/><span class="kw">then </span>
<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">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="c9">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><a NAME="E4:32_8_1_1_1_1_1"/><span class="kw">then </span>
<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">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="c9">ei</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E5:32_8_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A81</font></span>: 
<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">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:32_8_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">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E7:32_8_1_1_1_1_1"/><span class="kw">then </span>
<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">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:32_8_1_1_1_1_1"/><span class="kw">then </span>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E18:32_8"><span class="lab"><font color="Green" title="E60">A59</font></span></a>, <a class="txt" href="#E4:32_8_1_1"><span class="lab"><font color="Green" title="E66">A65</font></span></a>, <a class="txt" href="#E1:32_8_1_1_1_1_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="txt" href="#E5:32_8_1_1_1_1_1"><span class="lab"><font color="Green" title="E82">A81</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E9:32_8_1_1_1_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="c9">ei</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:32"><span class="lab"><font color="Green" title="E48">A43</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_8_1_1_1_1_2"/><span class="lab"><font color="Green" title="E81">A82</font></span>: 
( <font color="Maroon" title="c21">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="c22">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="c9">ei</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="c9">ei</font></span><br/><div class="add"><a NAME="E2:32_8_1_1_1_1_2"/>
<font color="Maroon" title="c25">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="32_8_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_8_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">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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:32_8_1_1"><span class="lab"><font color="Green" title="E78">A77</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E2:32_8_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">ni</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">ei</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/>
<a NAME="E3:32_8_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">ni</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">ei</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:32_8_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A83</font></span>: 
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:32_8_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E83">A84</font></span>: 
<font color="Maroon" title="c25">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c26">ni</font> = <font color="Maroon" title="c25">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_8_1_1"><span class="lab"><font color="Green" title="E67">A66</font></span></a></span>;<br/>
<a NAME="E7:32_8_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> <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="c26">ni</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_8_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E83">A84</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E8:32_8_1_1_1_1_2_1"/><span class="kw">then </span>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E16:32_8"><span class="lab"><font color="Green" title="E59">A58</font></span></a>, <a class="txt" href="#E25:32_8_1_1"><span class="lab"><font color="Green" title="E79">A78</font></span></a>, <a class="txt" href="#E1:32_8_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A82</font></span></a>, <a class="txt" href="#E4:32_8_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A83</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:32_8_1_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:32"><span class="lab"><font color="Green" title="E48">A43</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:32_8_1_1_1_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="c9">ei</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E18:32_8"><span class="lab"><font color="Green" title="E60">A59</font></span></a>, <a class="txt" href="#E6:32_8_1_1"><span class="lab"><font color="Green" title="E67">A66</font></span></a>, <a class="txt" href="#E3:32_8_1_1"><span class="lab"><font color="Green" title="E65">A64</font></span></a>, <a class="txt" href="#E1:32_8_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A82</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c26">im</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E28:32_8_1_1"/><span class="lab"><font color="Green" title="E81">A85</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">im</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E29:32_8_1_1"/><span class="lab"><font color="Green" title="E82">A86</font></span>: 
( <font color="Maroon" title="c22">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="c9">ei</font> or <font color="Maroon" title="c22">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:32"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="txt" href="#E20:32_8_1_1"><span class="lab"><font color="Green" title="E76">A75</font></span></a>, <a class="txt" href="#E19:32_8_1_1"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="txt" href="#E12:32_8_1_1"><span class="lab"><font color="Green" title="E70">A69</font></span></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#T22" title="ZFMISC_1:th.22">ZFMISC_1:22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">im</font> = <font color="Maroon" title="c26">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E31:32_8_1_1"/><span class="lab"><font color="Green" title="E83">A87</font></span>: 
<font color="Maroon" title="c27">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="c9">ei</font></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_8_1_1"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="txt" href="#E28:32_8_1_1"><span class="lab"><font color="Green" title="E81">A85</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E32:32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A88</font></span>: 
<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="c9">ei</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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="c27">im</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E28:32_8_1_1"><span class="lab"><font color="Green" title="E81">A85</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><a NAME="E33:32_8_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="c9">ei</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_8_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_8_1_1_2_1"/>
( ( <font color="Maroon" title="c18">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="c9">ei</font> &amp; <font color="Maroon" title="c19">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="c18">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="c19">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="c9">ei</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:32_8"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="txt" href="#E12:32_8"><span class="lab"><font color="Green" title="E57">A56</font></span></a>, <a class="txt" href="#E1:32_8_1_1"><span class="lab"><font color="Green" title="E63">A62</font></span></a>, <a class="txt" href="#E21:32_8_1_1"><span class="lab"><font color="Green" title="E77">A76</font></span></a>, <a class="txt" href="#E29:32_8_1_1"><span class="lab"><font color="Green" title="E82">A86</font></span></a>, <a class="ref" href="glib_000.html#T15" title="GLIB_000:th.15">GLIB_000:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_8_1_1_2_1_1"/><span class="lab"><font color="Green" title="E85">A89</font></span>: 
( <font color="Maroon" title="c18">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="c9">ei</font> &amp; <font color="Maroon" title="c19">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c9">ei</font></span><br/><div class="add"><a NAME="E2:32_8_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">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="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_8"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:32_8_1_1_2_1_1"/><span class="kw">then </span>
<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">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="c9">ei</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><a NAME="E4:32_8_1_1_2_1_1"/><span class="kw">then </span>
<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">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="c9">ei</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E5:32_8_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A90</font></span>: 
<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">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:32_8_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">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E7:32_8_1_1_2_1_1"/><span class="kw">then </span>
<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">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:32_8_1_1_2_1_1"/><span class="kw">then </span>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_8"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="txt" href="#E15:32_8_1_1"><span class="lab"><font color="Green" title="E73">A72</font></span></a>, <a class="txt" href="#E1:32_8_1_1_2_1_1"><span class="lab"><font color="Green" title="E85">A89</font></span></a>, <a class="txt" href="#E5:32_8_1_1_2_1_1"><span class="lab"><font color="Green" title="E86">A90</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E9:32_8_1_1_2_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="c9">ei</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_8_1_1_2_1_2"/><span class="lab"><font color="Green" title="E85">A91</font></span>: 
( <font color="Maroon" title="c18">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="c19">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="c9">ei</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="c9">ei</font></span><br/><div class="add"><a NAME="E2:32_8_1_1_2_1_2"/>
<font color="Maroon" title="c27">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="32_8_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_8_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">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="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:32_8_1_1"><span class="lab"><font color="Green" title="E83">A87</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E2:32_8_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">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="c9">ei</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/>
<a NAME="E3:32_8_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">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="c9">ei</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:32_8_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A92</font></span>: 
<font color="Maroon" title="c7">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:32_8_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E87">A93</font></span>: 
<font color="Maroon" title="c27">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c28">im</font> = <font color="Maroon" title="c27">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:32_8_1_1"><span class="lab"><font color="Green" title="E81">A85</font></span></a></span>;<br/>
<a NAME="E7:32_8_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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_8_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E87">A93</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E8:32_8_1_1_2_1_2_1"/><span class="kw">then </span>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_8"><span class="lab"><font color="Green" title="E55">A54</font></span></a>, <a class="txt" href="#E32:32_8_1_1"><span class="lab"><font color="Green" title="E84">A88</font></span></a>, <a class="txt" href="#E1:32_8_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A91</font></span></a>, <a class="txt" href="#E4:32_8_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E86">A92</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:32_8_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:32_8_1_1_2_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="c9">ei</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E11:32_8"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="txt" href="#E28:32_8_1_1"><span class="lab"><font color="Green" title="E81">A85</font></span></a>, <a class="txt" href="#E14:32_8_1_1"><span class="lab"><font color="Green" title="E72">A71</font></span></a>, <a class="txt" href="#E1:32_8_1_1_2_1_2"><span class="lab"><font color="Green" title="E85">A91</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E34:32_8_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E34:32"><span class="lab"><font color="Green" title="E49">A44</font></span></a>, <a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E26:32_8_1_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_8_1_2"/>
<font color="Maroon" title="c21">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:32_8_1_2"/><span class="kw">then </span>
( <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="c9">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="c21">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="c9">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="c21">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_8"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="txt" href="#E16:32_8"><span class="lab"><font color="Green" title="E59">A58</font></span></a>, <a class="txt" href="#E18:32_8"><span class="lab"><font color="Green" title="E60">A59</font></span></a>, <a class="txt" href="#E20:32_8"><span class="lab"><font color="Green" title="E62">A61</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_8_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:32"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E36:32"/>
 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="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="glib_001.html#T129" title="GLIB_001:th.129">GLIB_001:129</a></span>;<br/>
<a NAME="E37:32"/><span class="kw">then </span>
<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="c9">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="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:32"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E25:32"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="txt" href="#E29:32"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E27:32"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="txt" href="#E31:32"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="txt" href="#E35:32"><span class="lab"><font color="Green" title="E50">A49</font></span></a>, <a class="ref" href="helly.html#T20" target="_self" title="HELLY:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:32"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
