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

<span class="kw">let </span><font color="Maroon" title="c1">d</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c1">d</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">G</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E1:80"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D2" target="_self" title="CHAIN_1:def.2">Def2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">G</font> be    <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c4">l</font>, <font color="Maroon" title="c5">r</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">d</font>, <font color="Maroon" title="c6">i0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font><span class="kw"> such that </span><br/><a NAME="E3:80"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c4">l</font>,<font color="Maroon" title="c5">r</font>
 <span class="kw">and </span><br/><a NAME="E4:80"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
( <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or ( <font color="Maroon" title="c1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> ) )
 <span class="kw">and </span><br/><a NAME="E5:80"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">and </span><br/><a NAME="E6:80"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> holds <br/>( <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T44" target="_self" title="CHAIN_1:th.44">Th44</a></span>;<br/>
<a NAME="E7:80"/>
 ex <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">A2</font> &amp; ( for <font color="Olive" title="b3">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> or <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A2</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:80_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds <br/> ( <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="chain_1.html#T36" target="_self" title="CHAIN_1:th.36">Th36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">A1</font> =  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c4">l</font>,<font color="Maroon" title="c4">l</font>, <font color="Maroon" title="c8">A2</font> =  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c5">r</font> as   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T39" target="_self" title="CHAIN_1:th.39">Th39</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">A1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c7">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">A2</font> &amp; ( for <font color="Olive" title="b2">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A2</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">A2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c8">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c7">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">A2</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> ) ) )</span><br/>

<a NAME="E3:80_1"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c7">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T28" target="_self" title="CHAIN_1:th.28">Th28</a></span>;<br/>
<a NAME="E4:80_1"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<font color="Maroon" title="c8">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T28" target="_self" title="CHAIN_1:th.28">Th28</a></span>;<br/>
<a NAME="E5:80_1"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="chain_1.html#T27" target="_self" title="CHAIN_1:th.27">Th27</a></span>;<br/>
<a NAME="E6:80_1"/><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="chain_1.html#T27" target="_self" title="CHAIN_1:th.27">Th27</a></span>;<br/>
<a NAME="E7:80_1"/><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_1"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E8:80_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:80_1"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:80_1"/>
( <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c8">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E4:80_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E7:80_1"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="chain_1.html#T54" target="_self" title="CHAIN_1:th.54">Th54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">A2</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:80_1"/>
<font color="Maroon" title="c7">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E4:80_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E11:80_1"/><span class="lab"><font color="Green" title="E65">A11</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c9">A</font> as   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">G</font> ;<br/>
<a NAME="E13:80_1"/><span class="lab"><font color="Green" title="E66">A12</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E11:80_1"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="ref" href="chain_1.html#T54" target="_self" title="CHAIN_1:th.54">Th54</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">d</font><span class="kw"> such that </span><br/><a NAME="E15:80_1"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E16:80_1"/><span class="lab"><font color="Green" title="E68">A14</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds  <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T38" target="_self" title="CHAIN_1:th.38">Th38</a></span>;<br/>
<a NAME="E17:80_1"/><span class="lab"><font color="Green" title="E69">A15</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_1"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="chain_1.html#T27" target="_self" title="CHAIN_1:th.27">Th27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_1_1"/>
( <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or ( <font color="Maroon" title="c1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E57">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1"/><span class="lab"><font color="Green" title="E70">A16</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E71">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/><a NAME="E1:80_1_1_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E2:80_1_1_1_1"/>
<font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:80"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E1:80_1_1_1"><span class="lab"><font color="Green" title="E70">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:80_1_1_1"/><span class="lab"><font color="Green" title="E72">A18</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80_1"><span class="lab"><font color="Green" title="E68">A14</font></span></a></span>;<br/><a NAME="E4:80_1_1_1"/><span class="lab"><font color="Green" title="E73">A19</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E13:80_1"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E17:80_1"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E2:80_1_1_1"><span class="lab"><font color="Green" title="E71">A17</font></span></a>, <a class="ref" href="chain_1.html#T25" target="_self" title="CHAIN_1:th.25">Th25</a></span>;<br/><a NAME="E5:80_1_1_1"/><span class="lab"><font color="Green" title="E74">A20</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E13:80_1"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E17:80_1"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E2:80_1_1_1"><span class="lab"><font color="Green" title="E71">A17</font></span></a>, <a class="ref" href="chain_1.html#T25" target="_self" title="CHAIN_1:th.25">Th25</a></span>;<br/><a NAME="E6:80_1_1_1"/><span class="lab"><font color="Green" title="E75">A21</font></span>: 
(  not <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or  not <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E3:80_1_1_1"><span class="lab"><font color="Green" title="E72">A18</font></span></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></span>;<br/><div><span class="lab"><font color="Green" title="E76">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> implies ( <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:80_1_1_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )</span><br/><a NAME="E2:80_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A23</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:80"><span class="lab"><font color="Green" title="E59">A5</font></span></a></span>;<br/><a NAME="E3:80_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A24</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E13:80_1"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E17:80_1"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E2:80_1_1_1"><span class="lab"><font color="Green" title="E71">A17</font></span></a>, <a class="ref" href="chain_1.html#T25" target="_self" title="CHAIN_1:th.25">Th25</a></span>;<br/><a NAME="E4:80_1_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E13:80_1"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E17:80_1"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E2:80_1_1_1"><span class="lab"><font color="Green" title="E71">A17</font></span></a>, <a class="ref" href="chain_1.html#T25" target="_self" title="CHAIN_1:th.25">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E5:80_1_1_1_2"/>
( <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1_2"><span class="lab"><font color="Green" title="E77">A23</font></span></a>, <a class="txt" href="#E3:80_1_1_1_2"><span class="lab"><font color="Green" title="E78">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E8:80_1_1_1"/>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_1_1_1_3_1"/>
( <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_1_1"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="txt" href="#E5:80_1_1_1"><span class="lab"><font color="Green" title="E74">A20</font></span></a>, <a class="txt" href="#E6:80_1_1_1"><span class="lab"><font color="Green" title="E75">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E77">A25</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c13">l</font> = <font color="Maroon" title="c4">l</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D4" target="_self" title="CHAIN_1:def.4">Def4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1_3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/><a NAME="E1:80_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E2:80_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_1_1"><span class="lab"><font color="Green" title="E76">A22</font></span></a>, <a class="txt" href="#E1:80_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E77">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:80_1_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/><span class="kw">hence </span><a NAME="E5:80_1_1_1_3_1_1"/>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_1"><span class="lab"><font color="Green" title="E67">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1_3_1_2"/><span class="lab"><font color="Green" title="E77">A26</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c5">r</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D4" target="_self" title="CHAIN_1:def.4">Def4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1_3_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/><a NAME="E1:80_1_1_1_3_1_2_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E2:80_1_1_1_3_1_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_1_1"><span class="lab"><font color="Green" title="E76">A22</font></span></a>, <a class="txt" href="#E1:80_1_1_1_3_1_2"><span class="lab"><font color="Green" title="E77">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:80_1_1_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/><span class="kw">hence </span><a NAME="E5:80_1_1_1_3_1_2"/>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_1"><span class="lab"><font color="Green" title="E67">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_2"/><span class="lab"><font color="Green" title="E70">A27</font></span>: 
( <font color="Maroon" title="c1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/><div class="add"><a NAME="E2:80_1_1_2"/><span class="lab"><font color="Green" title="E71">A28</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds  <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font></span><br/>

<a NAME="E1:80_1_1_2_1"/><span class="lab"><font color="Green" title="E72">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E2:80_1_1_2_1"/><span class="lab"><font color="Green" title="E73">A30</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E3:80_1_1_2_1"/><span class="lab"><font color="Green" title="E74">A31</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:80_1_1_2_1"/><span class="lab"><font color="Green" title="E75">A32</font></span>: 
<font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:80_1_1_2_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E1:80_1_1_2_1"><span class="lab"><font color="Green" title="E72">A29</font></span></a>, <a class="txt" href="#E2:80_1_1_2_1"><span class="lab"><font color="Green" title="E73">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:80_1_1_2_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E3:80_1_1_2_1"><span class="lab"><font color="Green" title="E74">A31</font></span></a>, <a class="txt" href="#E4:80_1_1_2_1"><span class="lab"><font color="Green" title="E75">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c12">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font><span class="kw"> such that </span><br/><a NAME="E4:80_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i1</font>
 <span class="kw">and </span><br/><a NAME="E5:80_1_1_2"/><span class="lab"><font color="Green" title="E72">A33</font></span>: 
( <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i1</font> or <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E13:80_1"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E17:80_1"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E1:80_1_1_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="ref" href="chain_1.html#T26" target="_self" title="CHAIN_1:th.26">Th26</a></span>;<br/><a NAME="E6:80_1_1_2"/><span class="lab"><font color="Green" title="E73">A34</font></span>: 
<font color="Maroon" title="c12">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_2"><span class="lab"><font color="Green" title="E71">A28</font></span></a></span>;<br/><a NAME="E7:80_1_1_2"/><span class="lab"><font color="Green" title="E74">A35</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80_1"><span class="lab"><font color="Green" title="E68">A14</font></span></a></span>;<br/><a NAME="E8:80_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A36</font></span>: 
not <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E1:80_1_1_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></span>;<br/><a NAME="E9:80_1_1_2"/><span class="lab"><font color="Green" title="E76">A37</font></span>: 
not <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E1:80_1_1_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E7:80_1_1_2"><span class="lab"><font color="Green" title="E74">A35</font></span></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></span>;<br/><span class="kw">thus </span><a NAME="E10:80_1_1_2"/>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_1_1_2_2_1"/>
( <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_1_1_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="txt" href="#E6:80_1_1_2"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E8:80_1_1_2"><span class="lab"><font color="Green" title="E75">A36</font></span></a>, <a class="txt" href="#E9:80_1_1_2"><span class="lab"><font color="Green" title="E76">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E77">A38</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c5">r</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D4" target="_self" title="CHAIN_1:def.4">Def4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/><a NAME="E1:80_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_2"><span class="lab"><font color="Green" title="E71">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:80_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E77">A38</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:80_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/><span class="kw">hence </span><a NAME="E5:80_1_1_2_2_1_1"/>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_1"><span class="lab"><font color="Green" title="E67">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E77">A39</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c14">l</font> = <font color="Maroon" title="c4">l</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D4" target="_self" title="CHAIN_1:def.4">Def4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/><a NAME="E1:80_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_2"><span class="lab"><font color="Green" title="E71">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:80_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c13">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E77">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:80_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/><span class="kw">hence </span><a NAME="E5:80_1_1_2_2_1_2"/>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> or <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_1"><span class="lab"><font color="Green" title="E67">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:80"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" title="CHAIN_1:func.10">del</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T6" target="_self" title="CHAIN_1:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
