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

<b>let </b><font color="Maroon">kc</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">km</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f</font> <a href="fintopo4.html#R2">is_continuous</a> <font color="Maroon">kc</font>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">km</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span>
 ;<br/>

<a NAME="E3:11"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>assume </b><a NAME="E4:11"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>  holds not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Olive">p</font>,<font color="Maroon">km</font>
 ;<br/>

<a NAME="E5:11"/><i><font color="Green">E41</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4">FINTOPO4:def 4</a></i>;<br/>
<a NAME="E6:11"/><i><font color="Green">E42</font></i>: 
 <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font> is <a href="fin_topo.html#V1">filled</a>
 
<b>by </b><i><a class="ref" href="fintopo4.html#T18">FINTOPO4:18</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">j</font> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:11_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E2:11_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:11_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:11_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:11_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E7:11_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="fin_topo.html#D4">FIN_TOPO:def 4</a></i>;<br/><a NAME="E8:11_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,0
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E9:11_1"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,0 <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">km</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:11_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E8:11"/><b>then </b><i><font color="Green">E50</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">j</font>
 
;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<a NAME="E10:11"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:11"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 ;<br/>
<a NAME="E13:11"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:11"/><i><font color="Green">E59</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E15:11"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E16:11"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E17:11"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E18:11"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E19:11"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 or ( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pk</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_2"/>
( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 or ( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1"><b>suppose </b></a><a NAME="E1:11_2_1"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:11_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:11_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:11_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 holds <br/>1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">j</font>
 <b>by </b><i/>;<br/><a NAME="E5:11_2_1"/>
1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:11_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">j0</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:11_2_1"/>
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j0</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:11_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_2"><b>suppose </b></a><a NAME="E1:11_2_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:11_2_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E4:11_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">pkm</font> = <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:11_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:11_2_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, </i>;<br/><a NAME="E8:11_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><b>then reconsider </b><font color="Maroon">pj</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="txt" href="fintopo5.html#E1:11"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E3:11_2_2_1"/>
<font color="Maroon">pj</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pj</font>
 <b>by </b><i>, <a class="ref" href="fin_topo.html#D4">FIN_TOPO:def 4</a></i>;<br/><a NAME="E4:11_2_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">pj</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pj</font>,0
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E5:11_2_2_1"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pj</font>,0 <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pj</font>,<font color="Maroon">km</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:11_2_2_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><a NAME="E10:11_2_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E11:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:11_2_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E13:11_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E14:11_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E15:11_2_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E16:11_2_2"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E17:11_2_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">pkm</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="fintopo4.html#D3">FINTOPO4:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_2_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:11_2_2_2"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:11_2_2_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E2:11"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E4:11_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="fintopo5.html#T3" target="_self">Th3</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E5:11_2_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pkm</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, <a class="txt" href="fintopo5.html#E2:11"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E6:11_2_2_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pkm</font>,0
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E7:11_2_2_2"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pkm</font>,0 <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pkm</font>,<font color="Maroon">km</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:11_2_2_2"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><a NAME="E19:11_2_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E20:11_2_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E21:11_2_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">jn</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E23:11_2_2"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="fintopo4.html#D3">FINTOPO4:def 3</a>, <a class="txt" href="fintopo5.html#E2:11"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E24:11_2_2"/><i><font color="Green">E97</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jn</font> &amp; <font color="Maroon">jn</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E25:11_2_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">jn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><a NAME="E26:11_2_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E27:11_2_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">jn</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E28:11_2_2"/>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pkm</font>,<font color="Maroon">km</font>
 <b>by </b><i>, </i>;<br/><a NAME="E29:11_2_2"/><b>then </b>
not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E30:11_2_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/><a NAME="E31:11_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, </i>;<br/><a NAME="E32:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E33:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="fintopo5.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E34:11_2_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="fintopo5.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E35:11_2_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">jn</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E36:11_2_2"/>
not <font color="Maroon">jn</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pk</font>,<font color="Maroon">km</font>
 <b>by </b><i/>;<br/><a NAME="E37:11_2_2"/><b>then </b>
not  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="fintopo5.html#E5:11"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E38:11_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="fintopo5.html#T4" target="_self">Th4</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E39:11_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/><a NAME="E40:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E41:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E42:11_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E43:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">km</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">km</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E44:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E45:11_2_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E46:11_2_2"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/><a NAME="E47:11_2_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E48:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E6:11"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E49:11_2_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">kc</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>reconsider </b><font color="Maroon">pfk</font> = <font color="Maroon">jn</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="txt" href="fintopo5.html#E4:11"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E51:11_2_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pk</font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pfk</font>,<font color="Maroon">kc</font>
 <b>by </b><i>, <a class="ref" href="fintopo4.html#D2">FINTOPO4:def 2</a></i>;<br/><a NAME="E52:11_2_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:11"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E53:11_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, , <a class="txt" href="fintopo5.html#E1:11"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E54:11_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E55:11_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">pk</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="fintopo5.html#E3:11"><i><font color="Green">E35</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E56:11_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pk</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><a NAME="E57:11_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pk</font>,0
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E58:11_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pk</font>,0</span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, </i>;<br/><a NAME="E59:11_2_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pfk</font>,<font color="Maroon">kc</font>
 <b>by </b><i>, </i>;<br/><a NAME="E60:11_2_2"/>
<font color="Maroon">jn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E61:11_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E62:11_2_2"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E63:11_2_2"/><i><font color="Green">E123</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span>
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/><a NAME="E64:11_2_2"/>
<font color="Maroon">kc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">kc</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E65:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">kc</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E66:11_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">kc</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E67:11_2_2"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
