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

<span class="kw">let </span><font color="Maroon" title="c1">d9</font> be    <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">d</font> being 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> <br/> for <font color="Olive" title="b2">G</font> being   <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Olive" title="b1">d</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>for <font color="Olive" title="b3">A</font> being  <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">d9</font>,<font color="Olive" title="b2">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#K9" title="CHAIN_1:func.9">star</a> <font color="Olive" title="b3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="c2">d</font>  st <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>for <font color="Olive" title="b2">A</font> being  <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">d9</font>,<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#K9" title="CHAIN_1:func.9">star</a> <font color="Olive" title="b2">A</font></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">G</font> be    <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c2">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="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies for <font color="Olive" title="b1">A</font> being  <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">d9</font>,<font color="Maroon" title="c3">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#K9" title="CHAIN_1:func.9">star</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>





<span class="kw">assume </span><a NAME="E1:79"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">d9</font>,<font color="Maroon" title="c3">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#K9" title="CHAIN_1:func.9">star</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E2:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c1">d9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">d9</font>,<font color="Maroon" title="c3">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#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">l</font>, <font color="Maroon" title="c6">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="c2">d</font>, <font color="Maroon" title="c7">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="c2">d</font><span class="kw"> such that </span><br/><a NAME="E4:79"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<font color="Maroon" title="c4">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="c5">l</font>,<font color="Maroon" title="c6">r</font>
 <span class="kw">and </span><br/><a NAME="E5:79"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">and </span><br/><a NAME="E6:79"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">and </span><br/><a NAME="E7:79"/><span class="lab"><font color="Green" title="E59">A6</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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">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="#E1:79"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="chain_1.html#T42" target="_self" title="CHAIN_1:th.42">Th42</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E60">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="c2">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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><a NAME="E1:79_1"/>
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E2:79_1"/>
<font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E59">A6</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="E9:79"/>
 ex <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b2">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b1">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B2</font> &amp; ( for <font color="Olive" title="b3">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> or <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B1</font> or <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B2</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:79_2"/>
 ex <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">r1</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="c2">d</font> st <br/>( <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="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b3">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="c2">d</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) ) ) or for <font color="Olive" title="b3">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="c2">d</font> holds <br/> ( <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <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="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c8">l1i0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:79_2_1"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">l1i0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="chain_1.html#T20" target="_self" title="CHAIN_1:th.20">Th20</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_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="E1:79_2_1_1"/>
( <font color="Maroon" title="c8">l1i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> or <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1i0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_2_1"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
<font color="Maroon" title="c8">l1i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</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">l1</font>, <font color="Olive" title="b2">r1</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="c2">d</font> st <br/>( <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="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b3">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="c2">d</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) ) ) or for <font color="Olive" title="b3">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="c2">d</font> holds <br/> ( <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <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="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) ) )</span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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="c2">d</font>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l1i0</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 ) );<br/><a NAME="E2:79_2_1_1_1"/><span class="lab"><font color="Green" title="E63">A10</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="c2">d</font>  ex <font color="Olive" title="b2">li</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">li</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c2">d</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">li</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Olive" title="b1">li</font>]</span><br/>

<a NAME="E1:79_2_1_1_1_1"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E2:79_2_1_1_1_1"/>
 ex <font color="Olive" title="b1">li</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Olive" title="b1">li</font>]
 ; <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="c9">l1</font> being   <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="c2">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:79_2_1_1_1"/><span class="lab"><font color="Green" title="E64">A11</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="c2">d</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:79_2_1_1_1"><span class="lab"><font color="Green" title="E63">A10</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c10">l1</font> = <font color="Maroon" title="c9">l1</font> as    <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="c2">d</font> <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/><span class="kw">take </span>
<font color="Maroon" title="c10">l1</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">r1</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="c2">d</font> st <br/>( <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="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Maroon" title="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) ) ) or for <font color="Olive" title="b2">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="c2">d</font> holds <br/> ( <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <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="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c6">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c10">l1</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">l</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="c6">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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E6:79_2_1_1_1"/>
( <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="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c10">l1</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">l</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="c6">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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c10">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">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="#E5:79"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E2:79_2_1"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:79_2_1_1_1"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="txt" href="#E4:79_2_1_1_1"><span class="lab"><font color="Green" title="E64">A11</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="79_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1i0</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">l1</font>, <font color="Olive" title="b2">r1</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="c2">d</font> st <br/>( <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="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b3">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="c2">d</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) ) ) or for <font color="Olive" title="b3">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="c2">d</font> holds <br/> ( <font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <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="Olive" title="b1">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c9">l1</font>, <font color="Maroon" title="c10">r1</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="c2">d</font><span class="kw"> such that </span><br/><a NAME="E3:79_2_1_1_2"/>
 <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c9">l1</font>,<font color="Maroon" title="c10">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K8" title="CHAIN_1:func.8">infinite-cell</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">and </span><br/><a NAME="E4:79_2_1_1_2"/><span class="lab"><font color="Green" title="E63">A13</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="c2">d</font> holds <br/> ( <font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">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#D11" target="_self" title="CHAIN_1:def.11">Def11</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">l1</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">r1</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="c2">d</font> st <br/>( <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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) ) ) or for <font color="Olive" title="b2">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="c2">d</font> holds <br/> ( <font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c10">r1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c9">l1</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">l</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="c10">r1</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )</span><br/><a NAME="E5:79_2_1_1_2"/><span class="lab"><font color="Green" title="E64">A14</font></span>: 
<font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2_1_1_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a></span>;<br/><a NAME="E6:79_2_1_1_2"/>
<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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2_1_1_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:79_2_1_1_2"/>
( <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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c9">l1</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">l</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="c10">r1</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c9">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">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="#E2:79_2_1"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:79_2_1_1_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E4:79_2_1_1_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="txt" href="#E5:79_2_1_1_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="chain_1.html#T23" target="_self" title="CHAIN_1:th.23">Th23</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">l1</font>, <font color="Maroon" title="c9">r1</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="c2">d</font><span class="kw"> such that </span><br/><a NAME="E3:79_2"/><span class="lab"><font color="Green" title="E61">A15</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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">and </span><br/><a NAME="E4:79_2"/><span class="lab"><font color="Green" title="E62">A16</font></span>: 
<font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">and </span><br/><a NAME="E5:79_2"/><span class="lab"><font color="Green" title="E63">A17</font></span>: 
( ( <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c8">l1</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">l</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="c9">r1</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/>
<div><span class="lab"><font color="Green" title="E64">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c2">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font></span><br/><a NAME="E1:79_2_2"/><span class="lab"><font color="Green" title="E65">A19</font></span>: 
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> implies <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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a></span>;<br/><a NAME="E2:79_2_2"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:79_2_2"/>
<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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E5:79_2"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E1:79_2_2"><span class="lab"><font color="Green" title="E65">A19</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="E7:79_2"/><span class="lab"><font color="Green" title="E65">A20</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="c2">d</font>  holds <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_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="E1:79_2_3_1"/>
( ( <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c8">l1</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">l</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="c9">r1</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79_2"><span class="lab"><font color="Green" title="E63">A17</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_3_1_1"/><span class="lab"><font color="Green" title="E66">A21</font></span>: 
( <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c8">l1</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">l</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="c9">r1</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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">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="c2">d</font>  holds <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c2">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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span><br/><a NAME="E1:79_2_3_1_1_1"/><span class="lab"><font color="Green" title="E67">A22</font></span>: 
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> implies <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a></span>;<br/><a NAME="E2:79_2_3_1_1_1"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:79_2_3_1_1_1"/>
<font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_3_1_1"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E1:79_2_3_1_1_1"><span class="lab"><font color="Green" title="E67">A22</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><span class="kw">hence </span><a NAME="E3:79_2_3_1_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="c2">d</font>  holds <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 ; <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="79_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_3_1_2"/>
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="c2">d</font> holds <br/> ( <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</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">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="c2">d</font>  holds <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_2_3_1_2"/>
( 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="c2">d</font>  holds <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 ; <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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">B1</font> =  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c8">l1</font>,<font color="Maroon" title="c9">r1</font> as   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_2"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="ref" href="chain_1.html#T41" target="_self" title="CHAIN_1:th.41">Th41</a></span>;<br/>
<a NAME="E9:79_2"/>
 ex <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">r2</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="c2">d</font> st <br/>( <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="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b3">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="c2">d</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) ) ) or for <font color="Olive" title="b3">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="c2">d</font> holds <br/> ( <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <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="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c11">r2i0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:79_2_4"/><span class="lab"><font color="Green" title="E66">A23</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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<font color="Maroon" title="c11">r2i0</font></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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="chain_1.html#T19" target="_self" title="CHAIN_1:th.19">Th19</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_4_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="E1:79_2_4_1"/>
( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r2i0</font> or <font color="Maroon" title="c11">r2i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_2_4"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_4_1_1"/><span class="lab"><font color="Green" title="E67">A24</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r2i0</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">l2</font>, <font color="Olive" title="b2">r2</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="c2">d</font> st <br/>( <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="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b3">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="c2">d</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) ) ) or for <font color="Olive" title="b3">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="c2">d</font> holds <br/> ( <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <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="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) ) )</span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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="c2">d</font>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r2i0</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 ) );<br/><a NAME="E2:79_2_4_1_1"/><span class="lab"><font color="Green" title="E68">A25</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="c2">d</font>  ex <font color="Olive" title="b2">ri</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">ri</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_4_1_1_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="c2">d</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">ri</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">i</font>,<font color="Olive" title="b1">ri</font>]</span><br/>

<a NAME="E1:79_2_4_1_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c7">i0</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E2:79_2_4_1_1_1"/>
 ex <font color="Olive" title="b1">ri</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">i</font>,<font color="Olive" title="b1">ri</font>]
 ; <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">r2</font> being   <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="c2">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:79_2_4_1_1"/><span class="lab"><font color="Green" title="E69">A26</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="c2">d</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:79_2_4_1_1"><span class="lab"><font color="Green" title="E68">A25</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c13">r2</font> = <font color="Maroon" title="c12">r2</font> as    <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="c2">d</font> <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/><span class="kw">take </span>
<font color="Maroon" title="c5">l</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">r2</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="c2">d</font> st <br/>( <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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) ) ) or for <font color="Olive" title="b2">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="c2">d</font> holds <br/> ( <font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c13">r2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c5">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">l</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="c13">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E6:79_2_4_1_1"/>
( <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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c5">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">l</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="c13">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">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="#E2:79_2_4"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="txt" href="#E1:79_2_4_1_1"><span class="lab"><font color="Green" title="E67">A24</font></span></a>, <a class="txt" href="#E4:79_2_4_1_1"><span class="lab"><font color="Green" title="E69">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_4_1_2"/><span class="lab"><font color="Green" title="E67">A27</font></span>: 
<font color="Maroon" title="c11">r2i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</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">l2</font>, <font color="Olive" title="b2">r2</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="c2">d</font> st <br/>( <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="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b3">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="c2">d</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) ) ) or for <font color="Olive" title="b3">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="c2">d</font> holds <br/> ( <font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <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="Olive" title="b1">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c12">l2</font>, <font color="Maroon" title="c13">r2</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="c2">d</font><span class="kw"> such that </span><br/><a NAME="E3:79_2_4_1_2"/>
 <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c12">l2</font>,<font color="Maroon" title="c13">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K8" title="CHAIN_1:func.8">infinite-cell</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">and </span><br/><a NAME="E4:79_2_4_1_2"/><span class="lab"><font color="Green" title="E68">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="c2">d</font> holds <br/> ( <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">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#D11" target="_self" title="CHAIN_1:def.11">Def11</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c12">l2</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">r2</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="c2">d</font> st <br/>( <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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> holds <br/>( <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) ) ) or for <font color="Olive" title="b2">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="c2">d</font> holds <br/> ( <font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c13">r2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c12">l2</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">l</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="c13">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )</span><br/><a NAME="E5:79_2_4_1_2"/><span class="lab"><font color="Green" title="E69">A29</font></span>: 
<font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2_4_1_2"><span class="lab"><font color="Green" title="E68">A28</font></span></a></span>;<br/><a NAME="E6:79_2_4_1_2"/>
<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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2_4_1_2"><span class="lab"><font color="Green" title="E68">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:79_2_4_1_2"/>
( <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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( ( <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c12">l2</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">l</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="c13">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c12">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">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="#E2:79_2_4"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="txt" href="#E1:79_2_4_1_2"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="txt" href="#E4:79_2_4_1_2"><span class="lab"><font color="Green" title="E68">A28</font></span></a>, <a class="txt" href="#E5:79_2_4_1_2"><span class="lab"><font color="Green" title="E69">A29</font></span></a>, <a class="ref" href="chain_1.html#T23" target="_self" title="CHAIN_1:th.23">Th23</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">l2</font>, <font color="Maroon" title="c12">r2</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="c2">d</font><span class="kw"> such that </span><br/><a NAME="E11:79_2"/><span class="lab"><font color="Green" title="E66">A30</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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">and </span><br/><a NAME="E12:79_2"/><span class="lab"><font color="Green" title="E67">A31</font></span>: 
<font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">and </span><br/><a NAME="E13:79_2"/><span class="lab"><font color="Green" title="E68">A32</font></span>: 
( ( <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c11">l2</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">l</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="c12">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/>
<div><span class="lab"><font color="Green" title="E69">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">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="c2">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c13">i</font></span><br/><a NAME="E1:79_2_5"/><span class="lab"><font color="Green" title="E70">A34</font></span>: 
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> implies <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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c13">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a></span>;<br/><a NAME="E2:79_2_5"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:79_2_5"/>
<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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:79_2"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E13:79_2"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="#E1:79_2_5"><span class="lab"><font color="Green" title="E70">A34</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="E15:79_2"/>
( 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="c2">d</font>  holds <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_6"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_2_6_1"/>
( ( <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c11">l2</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">l</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="c12">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:79_2"><span class="lab"><font color="Green" title="E68">A32</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_6_1_1"/><span class="lab"><font color="Green" title="E70">A35</font></span>: 
( <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c11">l2</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">l</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="c12">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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">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="c2">d</font>  holds <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_6_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">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="c2">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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span><br/><a NAME="E1:79_2_6_1_1_1"/><span class="lab"><font color="Green" title="E71">A36</font></span>: 
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> implies <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a></span>;<br/><a NAME="E2:79_2_6_1_1_1"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:79_2_6_1_1_1"/>
<font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_6_1_1"><span class="lab"><font color="Green" title="E70">A35</font></span></a>, <a class="txt" href="#E1:79_2_6_1_1_1"><span class="lab"><font color="Green" title="E71">A36</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><span class="kw">hence </span><a NAME="E3:79_2_6_1_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="c2">d</font>  holds <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 ; <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="79_2_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_6_1_2"/>
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="c2">d</font> holds <br/> ( <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</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">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="c2">d</font>  holds <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_2_6_1_2"/>
( 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="c2">d</font>  holds <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 ; <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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">B2</font> =  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c11">l2</font>,<font color="Maroon" title="c12">r2</font> as   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:79_2"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="ref" href="chain_1.html#T41" target="_self" title="CHAIN_1:th.41">Th41</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">B1</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">B2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c10">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B2</font> &amp; ( for <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> or <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B2</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">B2</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="c10">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c13">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c10">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">B2</font> &amp; ( for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> ) ) )</span><br/>

<a NAME="E17:79_2"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_7"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_7_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="E1:79_2_7_1"/>
( ( <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c8">l1</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">l</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="c9">r1</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79_2"><span class="lab"><font color="Green" title="E63">A17</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_7_1_1"/><span class="lab"><font color="Green" title="E70">A37</font></span>: 
( <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c8">l1</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">l</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="c9">r1</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E71">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_7_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="c2">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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/><a NAME="E1:79_2_7_1_1_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c7">i0</font> &amp; <font color="Maroon" title="c8">l1</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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c9">r1</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="c6">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="#E1:79_2_7_1_1"><span class="lab"><font color="Green" title="E70">A37</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:79_2_7_1_1_1"/>
<font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r1</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="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E1:79_2_7_1_1"><span class="lab"><font color="Green" title="E70">A37</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="hs2(this)" href="javascript:()" title="79_2_7_1_1_2"><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="c2">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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/><a NAME="E1:79_2_7_1_1_2"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c7">i0</font> &amp; <font color="Maroon" title="c8">l1</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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c9">r1</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="c6">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="#E1:79_2_7_1_1"><span class="lab"><font color="Green" title="E70">A37</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:79_2_7_1_1_2"/>
( <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r1</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="#E5:79"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E2:79_2_7_1_1"><span class="lab"><font color="Green" title="E71">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><span class="kw">hence </span><a NAME="E4:79_2_7_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E2:79_2_7_1_1"><span class="lab"><font color="Green" title="E71">A38</font></span></a>, <a class="ref" href="chain_1.html#T29" target="_self" title="CHAIN_1:th.29">Th29</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="79_2_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_7_1_2"/>
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="c2">d</font> holds <br/> ( <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</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">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_2_7_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="chain_1.html#T31" target="_self" title="CHAIN_1:th.31">Th31</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>
<span class="kw">hence </span><a NAME="E18:79_2"/>
<font color="Maroon" title="c10">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E54">A1</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"> ( <font color="Maroon" title="c13">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c10">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">B2</font> &amp; ( for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> ) ) )</span><br/>

<a NAME="E19:79_2"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_8"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_8_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_2_8_1"/>
( ( <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c11">l2</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">l</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="c12">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) ) or 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="c2">d</font> holds <br/> ( <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:79_2"><span class="lab"><font color="Green" title="E68">A32</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_8_1_1"/><span class="lab"><font color="Green" title="E70">A39</font></span>: 
( <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; ( 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="c2">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="c7">i0</font> holds <br/>( <font color="Maroon" title="c11">l2</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">l</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="c12">r2</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="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B2</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E71">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_8_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="c2">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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/><a NAME="E1:79_2_8_1_1_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c7">i0</font> &amp; <font color="Maroon" title="c11">l2</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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c12">r2</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="c6">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="#E1:79_2_8_1_1"><span class="lab"><font color="Green" title="E70">A39</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:79_2_8_1_1_1"/>
<font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</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="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E1:79_2_8_1_1"><span class="lab"><font color="Green" title="E70">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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_8_1_1_2"><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="c2">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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/><a NAME="E1:79_2_8_1_1_2"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c7">i0</font> &amp; <font color="Maroon" title="c11">l2</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="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c12">r2</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="c6">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="#E1:79_2_8_1_1"><span class="lab"><font color="Green" title="E70">A39</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:79_2_8_1_1_2"/>
( <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r2</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="#E5:79"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E2:79_2_8_1_1"><span class="lab"><font color="Green" title="E71">A40</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><span class="kw">hence </span><a NAME="E4:79_2_8_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E2:79_2_8_1_1"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="ref" href="chain_1.html#T29" target="_self" title="CHAIN_1:th.29">Th29</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="79_2_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_8_1_2"/>
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="c2">d</font> holds <br/> ( <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <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="c11">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</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">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_2_8_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="ref" href="chain_1.html#T31" target="_self" title="CHAIN_1:th.31">Th31</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>
<span class="kw">hence </span><a NAME="E20:79_2"/>
<font color="Maroon" title="c13">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E54">A1</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"> ( <font color="Maroon" title="c10">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">B2</font> &amp; ( for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> ) ) )</span><br/>

<a NAME="E21:79_2"/><span class="lab"><font color="Green" title="E70">A41</font></span>: 
<font color="Maroon" title="c8">l1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">l2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E5:79_2"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a></span>;<br/>
<a NAME="E22:79_2"/>
( 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="c2">d</font> holds  <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c9">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79_2"><span class="lab"><font color="Green" title="E65">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:79_2"/>
<font color="Maroon" title="c10">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:79_2"><span class="lab"><font color="Green" title="E70">A41</font></span></a>, <a class="ref" href="chain_1.html#T32" target="_self" title="CHAIN_1:th.32">Th32</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">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">B</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="c14">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E24:79_2"/><span class="lab"><font color="Green" title="E71">A42</font></span>: 
<font color="Maroon" title="c14">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">B</font> = <font color="Maroon" title="c14">B</font> as   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c3">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E54">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">l9</font>, <font color="Maroon" title="c17">r9</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="c2">d</font><span class="kw"> such that </span><br/><a NAME="E27:79_2"/><span class="lab"><font color="Green" title="E72">A43</font></span>: 
<font color="Maroon" title="c15">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="c16">l9</font>,<font color="Maroon" title="c17">r9</font>
 <span class="kw">and </span><br/><a NAME="E28:79_2"/><span class="lab"><font color="Green" title="E73">A44</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="c2">d</font> holds  <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="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E29:79_2"/><span class="lab"><font color="Green" title="E74">A45</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="c2">d</font>  holds <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T40" target="_self" title="CHAIN_1:th.40">Th40</a></span>;<br/>
<a NAME="E30:79_2"/><span class="lab"><font color="Green" title="E75">A46</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="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:79_2"><span class="lab"><font color="Green" title="E73">A44</font></span></a></span>;<br/>
<a NAME="E31:79_2"/><span class="lab"><font color="Green" title="E76">A47</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:79_2"><span class="lab"><font color="Green" title="E71">A42</font></span></a>, <a class="ref" href="chain_1.html#T51" target="_self" title="CHAIN_1:th.51">Th51</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_9"><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="E1:79_2_9"/>
( 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="c2">d</font>  holds <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or 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="c2">d</font>  holds <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:79_2"><span class="lab"><font color="Green" title="E74">A45</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_9_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_9_1"/><span class="lab"><font color="Green" title="E77">A48</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="c2">d</font>  holds <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E78">A49</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_9_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">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="c2">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="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> implies ( <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:79_2_9_1_1"/><span class="lab"><font color="Green" title="E79">A50</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">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="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> )</span><br/><a NAME="E2:79_2_9_1_1"/>
<font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_9_1"><span class="lab"><font color="Green" title="E77">A48</font></span></a></span>;<br/><a NAME="E3:79_2_9_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> ) or ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> ) or ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E27:79_2"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="txt" href="#E31:79_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="ref" href="chain_1.html#T46" target="_self" title="CHAIN_1:th.46">Th46</a></span>;<br/><span class="kw">hence </span><a NAME="E4:79_2_9_1_1"/>
( <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E1:79_2_9_1_1"><span class="lab"><font color="Green" title="E79">A50</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><span class="kw">thus </span><a NAME="E3:79_2_9_1"/>
( <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</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="79_2_9_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:79_2_9_1_2"/><span class="lab"><font color="Green" title="E79">A51</font></span>: 
<font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_9_1"><span class="lab"><font color="Green" title="E77">A48</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_9_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_2_9_1_2_1"/>
( ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> ) or ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E27:79_2"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="txt" href="#E31:79_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2"><span class="lab"><font color="Green" title="E79">A51</font></span></a>, <a class="ref" href="chain_1.html#T46" target="_self" title="CHAIN_1:th.46">Th46</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_9_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_9_1_2_1_1"/><span class="lab"><font color="Green" title="E80">A52</font></span>: 
( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/><div class="add"><a NAME="E2:79_2_9_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A53</font></span>: 
<font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E30:79_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="ref" href="chain_1.html#T22" target="_self" title="CHAIN_1:th.22">Th22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">l9</font> = <font color="Maroon" title="c16">l9</font>, <font color="Maroon" title="c19">r9</font> = <font color="Maroon" title="c17">r9</font>, <font color="Maroon" title="c20">l1</font> = <font color="Maroon" title="c8">l1</font>, <font color="Maroon" title="c21">r1</font> = <font color="Maroon" title="c9">r1</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="c2">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><span class="lab"><font color="Green" title="E82">A54</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_9_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">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="c2">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="c18">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c19">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/><a NAME="E1:79_2_9_1_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A55</font></span>: 
<font color="Maroon" title="c20">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_9_1"><span class="lab"><font color="Green" title="E77">A48</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_1"><span class="lab"><font color="Green" title="E80">A52</font></span></a>, <a class="txt" href="#E2:79_2_9_1_2_1_1"><span class="lab"><font color="Green" title="E81">A53</font></span></a></span>;<br/><a NAME="E2:79_2_9_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c18">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c20">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E5:79_2"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E2:79_2_9_1"><span class="lab"><font color="Green" title="E78">A49</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:79_2_9_1_2_1_1_1"/>
<font color="Maroon" title="c18">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E30:79_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_1"><span class="lab"><font color="Green" title="E80">A52</font></span></a>, <a class="ref" href="chain_1.html#T22" target="_self" title="CHAIN_1:th.22">Th22</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="c19">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><a NAME="E4:79_2_9_1_2_1_1_1"/>
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c19">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c21">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E5:79_2"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E2:79_2_9_1"><span class="lab"><font color="Green" title="E78">A49</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A55</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:79_2_9_1_2_1_1_1"/>
<font color="Maroon" title="c19">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_1"><span class="lab"><font color="Green" title="E80">A52</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="E5:79_2_9_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">l9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">l1</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="E6:79_2_9_1_2_1_1"/>
( <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:79_2"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="txt" href="#E4:79_2_9_1_2_1_1"><span class="lab"><font color="Green" title="E82">A54</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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="79_2_9_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_9_1_2_1_2"/><span class="lab"><font color="Green" title="E80">A56</font></span>: 
( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/><div class="add"><a NAME="E2:79_2_9_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A57</font></span>: 
<font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:79_2"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E30:79_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="ref" href="chain_1.html#T21" target="_self" title="CHAIN_1:th.21">Th21</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">l9</font> = <font color="Maroon" title="c16">l9</font>, <font color="Maroon" title="c19">r9</font> = <font color="Maroon" title="c17">r9</font>, <font color="Maroon" title="c20">l2</font> = <font color="Maroon" title="c11">l2</font>, <font color="Maroon" title="c21">r2</font> = <font color="Maroon" title="c12">r2</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="c2">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><span class="lab"><font color="Green" title="E82">A58</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_9_1_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">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="c2">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="c19">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c18">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/><a NAME="E1:79_2_9_1_2_1_2_1"/><span class="lab"><font color="Green" title="E83">A59</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_9_1"><span class="lab"><font color="Green" title="E77">A48</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_2"><span class="lab"><font color="Green" title="E80">A56</font></span></a>, <a class="txt" href="#E2:79_2_9_1_2_1_2"><span class="lab"><font color="Green" title="E81">A57</font></span></a></span>;<br/><a NAME="E2:79_2_9_1_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c19">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c21">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E13:79_2"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="#E2:79_2_9_1"><span class="lab"><font color="Green" title="E78">A49</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:79_2_9_1_2_1_2_1"/>
<font color="Maroon" title="c19">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:79_2"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E30:79_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_2"><span class="lab"><font color="Green" title="E80">A56</font></span></a>, <a class="ref" href="chain_1.html#T21" target="_self" title="CHAIN_1:th.21">Th21</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="c18">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><a NAME="E4:79_2_9_1_2_1_2_1"/>
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0</font> or ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c18">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c20">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E13:79_2"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="#E2:79_2_9_1"><span class="lab"><font color="Green" title="E78">A49</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_2_1"><span class="lab"><font color="Green" title="E83">A59</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:79_2_9_1_2_1_2_1"/>
<font color="Maroon" title="c18">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E1:79_2_9_1_2_1_2"><span class="lab"><font color="Green" title="E80">A56</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="E5:79_2_9_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">l9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">l2</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="E6:79_2_9_1_2_1_2"/>
( <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:79_2"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="txt" href="#E4:79_2_9_1_2_1_2"><span class="lab"><font color="Green" title="E82">A58</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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="79_2_9_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_9_2"/><span class="lab"><font color="Green" title="E77">A60</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="c2">d</font>  holds <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c18">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="c2">d</font><span class="kw"> such that </span><br/><a NAME="E3:79_2_9_2"/><span class="lab"><font color="Green" title="E78">A61</font></span>: 
( ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> ) or ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E27:79_2"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="txt" href="#E31:79_2"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="ref" href="chain_1.html#T47" target="_self" title="CHAIN_1:th.47">Th47</a></span>;<br/><a NAME="E4:79_2_9_2"/><span class="lab"><font color="Green" title="E79">A62</font></span>: 
<font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E3:79_2_9_2"><span class="lab"><font color="Green" title="E78">A61</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E5:79_2_9_2"/>
( <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</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="79_2_9_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_9_2_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="E1:79_2_9_2_1_1"/>
( ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> ) or ( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2_9_2"><span class="lab"><font color="Green" title="E78">A61</font></span></a>, <a class="txt" href="#E4:79_2_9_2"><span class="lab"><font color="Green" title="E79">A62</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_9_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_9_2_1_1_1"/><span class="lab"><font color="Green" title="E80">A63</font></span>: 
( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/><div class="add"><a NAME="E2:79_2_9_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E30:79_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="ref" href="chain_1.html#T22" target="_self" title="CHAIN_1:th.22">Th22</a></span>;<br/><a NAME="E3:79_2_9_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K8" title="CHAIN_1:func.8">infinite-cell</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E5:79_2"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E1:79_2_9_2"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="txt" href="#E1:79_2_9_2_1_1_1"><span class="lab"><font color="Green" title="E80">A63</font></span></a>, <a class="ref" href="chain_1.html#T49" target="_self" title="CHAIN_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E4:79_2_9_2_1_1_1"/>
( <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:79_2"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="txt" href="#E1:79_2_9_2"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="ref" href="chain_1.html#T49" target="_self" title="CHAIN_1:th.49">Th49</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="79_2_9_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_9_2_1_1_2"/><span class="lab"><font color="Green" title="E80">A64</font></span>: 
( <font color="Maroon" title="c5">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )</span><br/><div class="add"><a NAME="E2:79_2_9_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:79_2"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E30:79_2"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="ref" href="chain_1.html#T21" target="_self" title="CHAIN_1:th.21">Th21</a></span>;<br/><a NAME="E3:79_2_9_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K8" title="CHAIN_1:func.8">infinite-cell</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_2"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E13:79_2"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="#E1:79_2_9_2"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="txt" href="#E1:79_2_9_2_1_1_2"><span class="lab"><font color="Green" title="E80">A64</font></span></a>, <a class="ref" href="chain_1.html#T49" target="_self" title="CHAIN_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E4:79_2_9_2_1_1_2"/>
( <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> or <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:79_2"><span class="lab"><font color="Green" title="E72">A43</font></span></a>, <a class="txt" href="#E1:79_2_9_2"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="ref" href="chain_1.html#T49" target="_self" title="CHAIN_1:th.49">Th49</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="E10:79"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K9" title="CHAIN_1:func.9">star</a> <font color="Maroon" title="c4">A</font></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>
