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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">d</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> or <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">d</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><b>suppose </b></a><a NAME="E1:91_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><a NAME="E2:91_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:91_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:91_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">C</font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font><br/> for <font color="Olive">l</font>, <font color="Olive">r</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>  st <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Olive">l</font>,<font color="Olive">r</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) holds <br/> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">r</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/>





<b>assume </b><a NAME="E1:91_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:91_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> ;<br/><b>set </b><font color="Maroon">BB</font> = <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>;<br/><a NAME="E1:91_1_1_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> iff ( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:91_1_1_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> iff ( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E10:91_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="chain_1.html#E3:91_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="chain_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><a NAME="E4:91_1_1_1_1"/><i><font color="Green">E63</font></i>: 
not  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, <a class="txt" href="chain_1.html#E4:91_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E5:91_1_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> 0 is <a href="abian.html#V1">even</a>
 ;<br/><b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:91_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E9:91_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, </i>;<br/><a NAME="E10:91_1_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E12:91_1_1_1_1"/>
<font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a></i>;<br/><b>then consider </b><font color="Maroon">Z</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span><b> such that </b><br/><a NAME="E14:91_1_1_1_1"/><i><font color="Green">E66</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E10:91_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, <a class="txt" href="chain_1.html#E1:91_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><b>consider </b><font color="Maroon">l'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E16:91_1_1_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l'</font>,<font color="Maroon">r'</font> &amp; (  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) or ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T6" target="_self">Th6</a>, <a class="ref" href="chain_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E17:91_1_1_1_1"/>
<font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , </i>;<br/><b>then consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span><b> such that </b><br/><a NAME="E19:91_1_1_1_1"/><i><font color="Green">E122</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 <b>by </b><i/>;<br/><a NAME="E20:91_1_1_1_1"/>
 ex <font color="Olive">B1</font>, <font color="Olive">B2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Olive">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">B2</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> or <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">B1</font> or <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">B2</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:91_1_1_1_1_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  st ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) holds <br/>( <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) ) &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  st not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) holds <br/>( <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) ) )
 
<b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , , , </i>;<br/>
<b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
2

;<br/>
<b>then consider </b><font color="Maroon">i1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">i2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:91_1_1_1_1_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> or <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> or <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> ) ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:91_1_1_1_1_2"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; not <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i1</font>, <font color="Maroon">i2</font> = <font color="Maroon">i2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Y1</font> = <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E7:91_1_1_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:91_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:91_1_1_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E10:91_1_1_1_1_2"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">xi</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<a NAME="E1:91_1_1_1_1_2_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:91_1_1_1_1_2_2"/>
 ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">xi</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">l1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:91_1_1_1_1_2"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E13:91_1_1_1_1_2"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">xi</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<a NAME="E1:91_1_1_1_1_2_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:91_1_1_1_1_2_3"/>
 ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Olive">xi</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">r1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E15:91_1_1_1_1_2"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">r1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">l1</font>, <font color="Maroon">r1</font> = <font color="Maroon">r1</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> <b>by </b><i/>;<br/>
<a NAME="E17:91_1_1_1_1_2"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<a NAME="E1:91_1_1_1_1_2_4"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<a NAME="E2:91_1_1_1_1_2_4"/><b>then </b>
( ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) or ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:91_1_1_1_1_2_4"/>
<font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:91_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<b>then </b><i><font color="Green">E134</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<a NAME="E20:91_1_1_1_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1_2_6_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_6_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_6_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_6_1_1"/><b>then </b>
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_6_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_6_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_6_1_2"/><i><font color="Green">E136</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_6_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:91_1_1_1_1_2_6_1_2"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:91_1_1_1_1_2_6_1_2"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B1</font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l1</font>,<font color="Maroon">r1</font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font> <b>by </b><i><a class="txt" href="chain_1.html#E14:91_1_1_1_1"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="chain_1.html#E1:91_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">Y2</font> = <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E22:91_1_1_1_1_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E23:91_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E24:91_1_1_1_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E25:91_1_1_1_1_2"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>,<font color="Olive">xi</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<a NAME="E1:91_1_1_1_1_2_7"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:91_1_1_1_1_2_7"/>
 ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">i</font>,<font color="Olive">xi</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">l2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E27:91_1_1_1_1_2"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>,<font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="chain_1.html#D5" target="_self">Def5</a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E28:91_1_1_1_1_2"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Olive">xi</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<a NAME="E1:91_1_1_1_1_2_8"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> )
 
;<br/>
<b>hence </b><a NAME="E2:91_1_1_1_1_2_8"/>
 ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">i</font>,<font color="Olive">xi</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">r2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E30:91_1_1_1_1_2"/><i><font color="Green">E182</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="chain_1.html#T10" target="_self">Th10</a>);<br/></i>
<b>reconsider </b><font color="Maroon">l2</font> = <font color="Maroon">l2</font>, <font color="Maroon">r2</font> = <font color="Maroon">r2</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> <b>by </b><i/>;<br/>
<a NAME="E32:91_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<b>then </b><i><font color="Green">E183</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<a NAME="E34:91_1_1_1_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1_2_10_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_10_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_10_1_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_10_1_1"/><b>then </b>
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_10_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, , <a class="ref" href="chain_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_10_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_10_1_2"/><i><font color="Green">E185</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_10_1_2"/><b>then </b><i><font color="Green">E186</font></i>: 
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:91_1_1_1_1_2_10_1_2"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_1_1_2_10_1_2"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T12" target="_self">Th12</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B2</font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l2</font>,<font color="Maroon">r2</font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font> <b>by </b><i><a class="txt" href="chain_1.html#E14:91_1_1_1_1"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="chain_1.html#E1:91_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">B1</font>
;<br/>

<b>take </b>
<font color="Maroon">B2</font>
;<br/>

<a NAME="E36:91_1_1_1_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1_2_11_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_11_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_11_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_11_1_1"/><b>then </b>
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_11_1_1"/>
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_11_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_11_1_2"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_11_1_2"/><b>then </b>
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_11_1_2"/>
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E37:91_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , </i>;<br/>
<b>hence </b><a NAME="E38:91_1_1_1_1_2"/>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a></i>;<br/>

<a NAME="E39:91_1_1_1_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1_2_12_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_12_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_12_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_12_1_1"/><b>then </b>
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_12_1_1"/>
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_12_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_12_1_2"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2_12_1_2"/><b>then </b>
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_12_1_2"/>
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E40:91_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , </i>;<br/>
<b>hence </b><a NAME="E41:91_1_1_1_1_2"/>
<font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_13"><b>now </b></a><div class="add"><a NAME="E1:91_1_1_1_1_2_13"/>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:91_1_1_1_1_2_13"/>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_13"/>
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:91_1_1_1_1_2_13"/>
( not <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:91_1_1_1_1_2_13"/>
not <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:91_1_1_1_1_2_13"/>
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T11" target="_self">Th11</a></i>;<br/><b>thus </b><a NAME="E7:91_1_1_1_1_2_13"/>
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E43:91_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">l1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l2</font> or <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r2</font> )
 
;<br/>
<b>hence </b><a NAME="E44:91_1_1_1_1_2"/>
<font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E45:91_1_1_1_1_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font> ;<br/>
<a NAME="E47:91_1_1_1_1_2"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, <a class="ref" href="chain_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon">l''</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r''</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E49:91_1_1_1_1_2"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l''</font>,<font color="Maroon">r''</font> &amp; (  ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) or ( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E14:91_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="chain_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E50:91_1_1_1_1_2"/>
<font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 
<b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , <a class="ref" href="chain_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span><b> such that </b><br/><a NAME="E52:91_1_1_1_1_2"/><i><font color="Green">E190</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E53:91_1_1_1_1_2"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  st ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) holds <br/>( <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) ) )
 
<b>by </b><i>, , , <a class="ref" href="chain_1.html#T13" target="_self">Th13</a>, <a class="ref" href="chain_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E54:91_1_1_1_1_2"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  st ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) holds <br/>( <font color="Maroon">l''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r''</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) ) )
 
<b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , <a class="ref" href="chain_1.html#T13" target="_self">Th13</a>, <a class="ref" href="chain_1.html#T14" target="_self">Th14</a>, </i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="chain_1.html#T14" target="_self">Th14</a>, <a class="ref" href="chain_1.html#T15" target="_self">Th15</a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
1

;<br/>
<b>then consider </b><font color="Maroon">i'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E57:91_1_1_1_1_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i'</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E58:91_1_1_1_1_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> ;<br/>
<a NAME="E60:91_1_1_1_1_2"/>
( <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="chain_1.html#D6" target="_self">Def6</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E61:91_1_1_1_1_2"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<i><font color="Green">E196</font></i>: <font color="Maroon">Y</font> = 
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="chain_1.html#T15" target="_self">Th15</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i'</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_16"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1_2_16"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_16_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_16_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">l''</font> = <font color="Maroon">l''</font>, <font color="Maroon">r''</font> = <font color="Maroon">r''</font>, <font color="Maroon">l1</font> = <font color="Maroon">l1</font>, <font color="Maroon">r1</font> = <font color="Maroon">r1</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_16_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:91_1_1_1_1_2_16_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><a NAME="E2:91_1_1_1_1_2_16_1_1"/><b>then </b>
( ( <font color="Maroon">l''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) or ( <font color="Maroon">l''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , , , , , <a class="ref" href="chain_1.html#T13" target="_self">Th13</a>, <a class="ref" href="chain_1.html#T14" target="_self">Th14</a>, <a class="ref" href="chain_1.html#T19" target="_self">Th19</a>, </i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_16_1_1"/>
( <font color="Maroon">l''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 ;<br/></div><b>end;</b></div><a NAME="E4:91_1_1_1_1_2_16_1"/><b>then </b>
( <font color="Maroon">l''</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r1</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:91_1_1_1_1_2_16_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2_16_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2_16_2"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i2</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">l''</font> = <font color="Maroon">l''</font>, <font color="Maroon">r''</font> = <font color="Maroon">r''</font>, <font color="Maroon">l2</font> = <font color="Maroon">l2</font>, <font color="Maroon">r2</font> = <font color="Maroon">r2</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_2_16_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:91_1_1_1_1_2_16_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><a NAME="E2:91_1_1_1_1_2_16_2_1"/><b>then </b>
( ( <font color="Maroon">l''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) or ( <font color="Maroon">l''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , , , , <a class="ref" href="chain_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="chain_1.html#T13" target="_self">Th13</a>, <a class="ref" href="chain_1.html#T14" target="_self">Th14</a>, <a class="ref" href="chain_1.html#T19" target="_self">Th19</a>, </i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_2_16_2_1"/>
( <font color="Maroon">l''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> )
 ;<br/></div><b>end;</b></div><a NAME="E4:91_1_1_1_1_2_16_2"/><b>then </b>
( <font color="Maroon">l''</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">l2</font> &amp; <font color="Maroon">r''</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:91_1_1_1_1_2_16_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E21:91_1_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E22:91_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, <a class="txt" href="chain_1.html#E4:91_1_1"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:91_1_1_1"/><b>then </b>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:91_1_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">C</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E5:91_1_1"/><i><font color="Green">E199</font></i>: 
for <font color="Olive">C1</font>, <font color="Olive">C2</font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>  st  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Olive">C1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font> &amp;  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Olive">C2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font> holds <br/> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2">(<span class="default"><font color="Olive">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">C2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C1</font> be   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>, <font color="Maroon">C2</font> be   <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:91_1_1_2"/><i><font color="Green">E200</font></i>: 
(  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font> &amp;  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font> )
 ;<br/>

<b>thus </b> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">C2</font></span>)</span></span>)</span> = 
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C1</font></span>)</span> <a href="chain_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="chain_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, </i>
<br/>.= 
 <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>

;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>] <b>means </b> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>;<br/><a NAME="E6:91_1_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="chain_1.html#K6" target="_self">0_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b> <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font></span>)</span></span>)</span> = 
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K6" target="_self">0_</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><a NAME="E7:91_1_1"/>
for <font color="Olive">A</font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font> holds   <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">A</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>;<br/>

<b>consider </b><font color="Maroon">l</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E2:91_1_1_4"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> &amp; (  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_4_1"/>
(  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 </span><b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_4_1_1"/>
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 ;<br/><div class="add"><a NAME="E2:91_1_1_4_1_1"/><b>then </b>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font>
 ;<br/><b>hence </b><a NAME="E3:91_1_1_4_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_4_1_2"/><i><font color="Green">E216</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) )
 ;<br/><div class="add"><a NAME="E2:91_1_1_4_1_2"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, <a class="txt" href="chain_1.html#E2:91_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>set </b><font color="Maroon">C</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a> ;<br/><a NAME="E3:91_1_1_4_1_2"/><i><font color="Green">E218</font></i>: 
for <font color="Olive">A</font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">A</font></span><a href="domain_1.html#K6">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A'</font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:91_1_1_4_1_2_1"/>
<font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:91_1_1_4_1_2_1"/><b>then </b>
not <font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:91_1_1_4_1_2_1"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">A'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">l'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r'</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E5:91_1_1_4_1_2_1"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l'</font>,<font color="Maroon">r'</font> &amp; (  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) or ( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_4_1_2_1_1"/>
(  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) or for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_4_1_2_1_1_1"/>
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 ;<br/><div class="add"><a NAME="E2:91_1_1_4_1_2_1_1_1"/><b>then </b>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font>
 ;<br/><b>hence </b><a NAME="E3:91_1_1_4_1_2_1_1_1"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A'</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_4_1_2_1_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_1_1_4_1_2_1_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A'</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="txt" href="chain_1.html#E2:91_1_1"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:91_1_1_4_1_2"/><i><font color="Green">E221</font></i>: 
for <font color="Olive">C1</font>, <font color="Olive">C2</font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>  st <font color="Olive">C1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">C2</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">C2</font>]
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E5:91_1_1_4_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K3">`</a> ]
 <b>from </b><i><a class="ref" href="chain_1.html#S4" target="_self">CHAIN_1:sch 4</a>(<a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, , );<br/></i><b>hence </b><a NAME="E6:91_1_1_4_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:91_1_1"/><b>then </b><i><font color="Green">E222</font></i>: 
for <font color="Olive">A</font> being  <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">A</font></span><a href="domain_1.html#K6">}</a></span>]
 ;<br/><a NAME="E9:91_1_1"/><i><font color="Green">E223</font></i>: 
for <font color="Olive">C1</font>, <font color="Olive">C2</font> being  <a href="chain_1.html#NM4">Chain</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>  st <font color="Olive">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Olive">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C1</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Olive">C2</font>]
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a></i>;<br/><b>thus </b><a NAME="E10:91_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>]
 <b>from </b><i><a class="ref" href="chain_1.html#S4" target="_self">CHAIN_1:sch 4</a>(<a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, <a class="ref" href="chain_1.html#D4" target="_self">Def4</a>);</i><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2"><b>suppose </b></a><a NAME="E1:91_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><a NAME="E2:91_1_2"/><b>then </b>
 <a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:91_1_2"/>
 <a href="chain_1.html#K11" target="_self">del</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K6" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
