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

<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">jn</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</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">p</font> be   <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>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn</font>
 ;<br/>

<a NAME="E2:10"/><i><font color="Green">E33</font></i>: 
 <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <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> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4">FINTOPO4:def 4</a></i>;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E36</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">p</font>,<font color="Maroon">k</font>
 ;<br/><a NAME="E2:10_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">j</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/><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>for <font color="Olive">j2</font>, <font color="Olive">jn2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p2</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>  st <font color="Olive">jn2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p2</font> &amp; <font color="Olive">j2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Olive">p2</font>,<font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E3:10_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jn2</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">p2</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,0 )
 ;<br/>

<a NAME="E2:10_1_1"/><i><font color="Green">E46</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jn2</font> &amp; <font color="Maroon">jn2</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="E3:10_1_1"/>
<font color="Maroon">j2</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="fintopo3.html#T47">FINTOPO3:47</a>, </i>;<br/>
<a NAME="E4:10_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">j2</font> <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"><font color="Maroon">jn2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/>
<a NAME="E5:10_1_1"/>
<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">jn2</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">jn2</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><font color="Maroon">jn2</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">jn2</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></i>;<br/>
<a NAME="E6:10_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1_2_1"/>
( <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:10_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a>, <a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:10_1_1_1_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:10_1_1_1_2_1_2"/>
<font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:10_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:10_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E6:10_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E7:10_1_1_1_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</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="E8:10_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E9:10_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">jn2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E10:10_1_1_1_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">jn2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E11:10_1_1_1_2_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3"><b>suppose </b></a><a NAME="E1:10_1_1_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1_3_1"/>
( <font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_3_1_1"/>
<font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:10_1_1_1_3_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:10_1_1_1_3_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_3_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:10_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E3:10_1_1_1_3_1_2"/>
<font color="Maroon">jn2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:10_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:10_1_1_1_3_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:10_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">i2</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">i2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i2</font>]
 ;<br/>

<b>let </b><font color="Maroon">j3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jn3</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">p2</font> be   <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>;<br/>



<b>assume </b><a NAME="E2:10_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">jn3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">j3</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">i2</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
<b>by </b><i><a class="ref" href="fintopo3.html#T48">FINTOPO3:48</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <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> :  ex <font color="Olive">y</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> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">i2</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> ) </span>}</span> 
<b>by </b><i><a class="ref" href="fin_topo.html#D14">FIN_TOPO:def 14</a></i>
;<br/>
<b>then consider </b><font color="Maroon">x</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><b> such that </b><br/><a NAME="E5:10_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j3</font> &amp;  ex <font color="Olive">y</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> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</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><b> such that </b><br/><a NAME="E7:10_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:10_1_2"/>
<font color="Maroon">y</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/>
<b>then reconsider </b><font color="Maroon">iy</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E10:10_1_2"/><i><font color="Green">E69</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn3</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">iy</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:10_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,0
 
<b>by </b><i>, <a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/>
<a NAME="E12:10_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">iy</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j3</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:10_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">jn3</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">iy</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">iy</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j3</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</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="fintopo5.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E14:10_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">jn3</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">iy</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">iy</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j3</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">jn3</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">iy</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">iy</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>
<b>hence </b><a NAME="E15:10_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn3</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j3</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:10_1"/>
for <font color="Olive">i3</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i3</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>hence </b><a NAME="E6:10_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<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>for <font color="Olive">j2</font>, <font color="Olive">jn2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p2</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>  st <font color="Olive">jn2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p2</font> &amp; <font color="Olive">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Olive">j2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Olive">p2</font>,<font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E2:10_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jn2</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">p2</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E2:10_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">jn2</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>, </i>;<br/>
<a NAME="E3:10_2_1"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</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="10_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_1_1_1"/>
( <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1"><b>case </b></a><a NAME="E1:10_2_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:10_2_1_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:10_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/><a NAME="E5:10_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:10_2_1_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_1_1_1_1_1"/>
( <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> or <font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 </span><b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_1_1_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_1_1_1"/>
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</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/><b>hence </b><a NAME="E3:10_2_1_1_1_1_1_1"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_1_1_1_1_1_2"/>
<font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_2_1_1_1_1_1_2"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_2"><b>case </b></a><a NAME="E1:10_2_1_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:10_2_1_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">jn2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">jn2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:10_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, </i>;<br/><a NAME="E5:10_2_1_1_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:10_2_1_1_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:10_2_1_1_1_2"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:10_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">jn2</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><font color="Maroon">jn2</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">jn2</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="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E6:10_2_1"/><b>then </b>
<font color="Maroon">j2</font> <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">jn2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="fintopo4.html#D3">FINTOPO4:def 3</a></i>;<br/>
<a NAME="E7:10_2_1"/><b>then </b>
<font color="Maroon">j2</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#D1">FIN_TOPO:def 1</a>, </i>;<br/>
<b>hence </b><a NAME="E8:10_2_1"/>
<font color="Maroon">j2</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/>


</div><b>end;</b></div><a NAME="E3:10_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">jj</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">jj</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">jj</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">jj</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_2_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">jj</font>]
 ;<br/>

<b>let </b><font color="Maroon">j2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jn2</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">p2</font> be   <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>;<br/>



<b>assume </b><a NAME="E2:10_2_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">j2</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="E4:10_2_2"/><i><font color="Green">E98</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</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="E5:10_2_2"/><i><font color="Green">E99</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jn2</font> &amp; <font color="Maroon">jn2</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><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_2_1_1"/>
( <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_2_1_1_1_1"/>
( <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">jn2</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 ;<br/><a NAME="E3:10_2_2_1_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:10_2_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:10_2_2_1_1_1_1_1"/>
 <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/><a NAME="E6:10_2_2_1_1_1_1_1"/><b>then </b>
( <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>,<font color="Maroon">jj</font> &amp; <font color="Maroon">x0</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/><b>hence </b><a NAME="E7:10_2_2_1_1_1_1_1"/>
 ex <font color="Olive">y</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> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_1_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E3:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</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/><a NAME="E4:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:10_2_2_1_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</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="E6:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:10_2_2_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E8:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">jj</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="txt" href="fintopo5.html#E1:10"><i><font color="Green">E24</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:10_2_2_1_1_1_1_2"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <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="E10:10_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E11:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj</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="E12:10_2_2_1_1_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:10_2_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">j2</font> <a href="nat_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/>;<br/><b>then reconsider </b><font color="Maroon">yj2</font> = <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</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/>;<br/><a NAME="E16:10_2_2_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E17:10_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font>
 <b>by </b><i>, , <a class="ref" href="fintopo5.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E18:10_2_2_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E19:10_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">yj2</font>,0
 <b>by </b><i>, </i>;<br/><a NAME="E20:10_2_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">yj2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">yj2</font> )
 <b>by </b><i>, <a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><b>hence </b><a NAME="E21:10_2_2_1_1_1_1_2"/>
 ex <font color="Olive">y</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> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_2_1_1_2"/>
<font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:10_2_2_1_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:10_2_2_1_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <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="E5:10_2_2_1_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</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="E6:10_2_2_1_1_2"/>
<font color="Maroon">j2</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="E7:10_2_2_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</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="E8:10_2_2_1_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:10_2_2_1_1_2"/>
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</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="E10:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon">j2</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">j2</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="E11:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, </i>;<br/><a NAME="E12:10_2_2_1_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">j2</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/>;<br/><b>then reconsider </b><font color="Maroon">pj21</font> = <font color="Maroon">j2</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/>;<br/><a NAME="E14:10_2_2_1_1_2"/><i><font color="Green">E138</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font>
 <b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E15:10_2_2_1_1_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><a NAME="E16:10_2_2_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font></span>)</span>
<b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">jn2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, </i>
<br/>.= 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>
;<br/><a NAME="E18:10_2_2_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">jn2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj</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/><a NAME="E19:10_2_2_1_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">j2</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">p2</font>,<font color="Maroon">jj</font>
 <b>by </b><i>, , <a class="txt" href="fintopo5.html#E1:10"><i><font color="Green">E24</font></i></a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j2</font></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E21:10_2_2_1_1_2"/><b>then </b>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">pj21</font>,0
 <b>by </b><i>, </i>;<br/><a NAME="E22:10_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">pj21</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pj21</font> )
 <b>by </b><i>, <a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><b>hence </b><a NAME="E23:10_2_2_1_1_2"/>
 ex <font color="Olive">y</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> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<span class="p1">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
<b>by </b><i><a class="ref" href="fintopo3.html#T48">FINTOPO3:48</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <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> :  ex <font color="Olive">y</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> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<font color="Maroon">jj</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font> ) </span>}</span> 
<b>by </b><i><a class="ref" href="fin_topo.html#D14">FIN_TOPO:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E8:10_2_2"/>
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p2</font>,<span class="p1">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:10_2"/>
for <font color="Olive">ii</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">ii</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>hence </b><a NAME="E5:10_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p</font>,<font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> iff ( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i/>;<br/>


</div>
