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<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>2</sub></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">a<sub>1</sub></font> &amp; ( for <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">xi</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">a<sub>1</sub></font> holds <br/><font color="Olive">xi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>2</sub></font> ) );<br/>
<a NAME="E1:71_1_1"/><i><font color="Green">E37</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">li</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">li</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</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="E3:71_1_1"/><i><font color="Green">E51</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">l</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">l</font> = <font color="Maroon">l</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/>
<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>2</sub></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">a<sub>1</sub></font> &amp; ( for <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">xi</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">a<sub>1</sub></font> holds <br/><font color="Olive">xi</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>2</sub></font> ) );<br/>
<a NAME="E5:71_1_1"/><i><font color="Green">E52</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">ri</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">ri</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</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="E7:71_1_1"/><i><font color="Green">E53</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">r</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="txt" href="chain_1.html#E3:71_1_1"><i><font color="Green">E51</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</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/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_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="E1:71_1_1_1"/>
( <font color="Maroon">l</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> &amp; <font color="Maroon">r</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#D3" target="_self">Def3</a>, <a class="txt" href="chain_1.html#E5:71_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E2:71_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<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">l</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#D3" target="_self">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">l</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>
 ;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x1</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> &amp; <font color="Maroon">x2</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> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><a NAME="E5:71_1_1_1_1"/>
( <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">x1</font> &amp; <font color="Maroon">x1</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">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">x2</font> &amp; <font color="Maroon">x2</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> )
 <b>by </b><i><a class="ref" href="chain_1.html#D3" target="_self">Def3</a>, <a class="txt" href="chain_1.html#E5:71_1_1"><i><font color="Green">E52</font></i></a>, </i>;<br/><a NAME="E6:71_1_1_1_1"/><b>then </b>
( <font color="Maroon">x1</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">x2</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> )
 <b>by </b><i><a class="ref" href="chain_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:71_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:71_1_1_1"/>
<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">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E9:71_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><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="E1:71_1_1_2"/>
( <font color="Maroon">l</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> &amp; <font color="Maroon">r</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> &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">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; ( for <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">xi</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> holds <br/>( not <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="Olive">xi</font> &amp; not <font color="Olive">xi</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#D3" target="_self">Def3</a>, <a class="txt" href="chain_1.html#E5:71_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="chain_1.html#E7:71_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E2:71_1_1_2"/>
( <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">l</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">l</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">r</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> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">d</font>,<font color="Maroon">G</font> <b>by </b><i><a class="txt" href="chain_1.html#E1:71_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

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

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

<b>thus </b><a NAME="E12:71_1_1"/>
( <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; ( 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#E9:71_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>


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