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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>] <b>means </b>(  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">a<sub>1</sub></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">a<sub>2</sub></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">a<sub>1</sub></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">a<sub>2</sub></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">a<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">a<sub>1</sub></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">a<sub>2</sub></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">a<sub>1</sub></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">a<sub>1</sub></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">a<sub>2</sub></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/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>) <b>-&gt; </b> non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font></span>)</span> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">CELLS</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>,<font color="Olive">r</font>) where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>,<font color="Olive">r</font>] </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">X</font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> as   <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>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></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="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  ex <font color="Olive">li</font>, <font color="Olive">ri</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">li</font>,<font color="Olive">ri</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">ri</font> &amp; <font color="Maroon">a<sub>2</sub></font> 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">a<sub>1</sub></font> ) ) or ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  ex <font color="Olive">li</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">li</font>,<font color="Olive">li</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">li</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> ) ) );<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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/><b>thus </b><a NAME="E1:49_1_1_1"/>
 ex <font color="Olive">lri</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Olive">lri</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">li</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">ri</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:49_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">li</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">ri</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">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</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> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">xi</font> holds <br/> not <font color="Olive">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> ) )
 <b>by </b><i/>;<br/><b>take </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K1">]</a></span>
;<br/><a NAME="E4:49_1_1_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></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><a class="ref" href="chain_1.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E5:49_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K1">]</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E1:49"><i><font color="Green">E37</font></i></a>, <a class="ref" href="chain_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_1_1_2"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">li</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i</font>;<br/><b>reconsider </b><font color="Maroon">li</font> = <font color="Maroon">li</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>reconsider </b><font color="Maroon">lri</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">li</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><b>take </b>
<font color="Maroon">lri</font>
;<br/><b>thus </b><a NAME="E5:49_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">lri</font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E1:49"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">lr</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E4:49_1_1"/><i><font color="Green">E60</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">lr</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#T1" target="_self">Th1</a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<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>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a> ;<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="E6:49_1_1"/><i><font color="Green">E61</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">l</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<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>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a> ;<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="E8:49_1_1"/><i><font color="Green">E63</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">r</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font>, <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">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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="E1:49_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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="domain_1.html#K2">`1</a>  &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="domain_1.html#K3">`2</a>  )
 <b>by </b><i><a class="ref" href="chain_1.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E2:49_1_1_2"/>
<font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_3"><b>now </b></a><div class="add"><b>take </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>;<br/><b>thus </b><a NAME="E1:49_1_1_3"/>
<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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">X</font> = <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E1:49_1_1_3_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><b>take </b><font color="Maroon">l</font> = <font color="Maroon">l</font>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E2:49_1_1_3_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>
 ;<br/><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="49_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_3_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_3_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">li</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">ri</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:49_1_1_3_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> &amp; <font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> 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><a class="txt" href="chain_1.html#E1:49"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E4:49_1_1_3_1_1_1"/>
<font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i/>;<br/><a NAME="E5:49_1_1_3_1_1_1"/><b>then </b>
( <font color="Maroon">li</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">ri</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="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:49_1_1_3_1_1_1"/>
( ( <font color="Maroon">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="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; <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> ) or ( not <font color="Maroon">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="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">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> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_3_1_1_2"/><i><font color="Green">E121</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">li</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:49_1_1_3_1_1_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">lr</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">li</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">li</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="txt" href="chain_1.html#E1:49"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E4:49_1_1_3_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">li</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i>, </i>;<br/><a NAME="E5:49_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">li</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">li</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="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:49_1_1_3_1_1_2"/>
( ( <font color="Maroon">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="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; <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> ) or ( not <font color="Maroon">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="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">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> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_3"/>
 ex <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 <br/>( <font color="Maroon">A</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; (  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="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> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">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="Olive">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="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Olive">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="Olive">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="Olive">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="Olive">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="Olive">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><b>end;</b></div>
<a NAME="E12:49_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>,<font color="Olive">r</font>) where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>,<font color="Olive">r</font>] </span>}</span> 
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">G</font> &amp; ( ( <font color="Maroon">a<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font> ) or ( <font color="Maroon">a<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font> ) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <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="Maroon">S<sub>3</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">l</font>,<font color="Olive">r</font>,<font color="Maroon">a<sub>2</sub></font>];<br/>
<a NAME="E13:49_1_1"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>,<font color="Olive">r</font>) where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>,<font color="Olive">r</font>] </span>}</span>  holds <br/> ex <font color="Olive">lr</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">A</font>,<font color="Olive">lr</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_4"><b>proof </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="E1:49_1_1_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>,<font color="Olive">r</font>) where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>,<font color="Olive">r</font>] </span>}</span> 
 ;<br/>

<b>then 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="E3:49_1_1_4"/><i><font color="Green">E125</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> ) ) ) ) )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_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="E1:49_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> <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> ) ) ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:49_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> <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> ) ) ) )
 ;<br/><div class="add"><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="E3:49_1_1_4_1_1"/><i><font color="Green">E126</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 <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> ) )
 ;<br/><b>take </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>
;<br/><b>take </b>
<font color="Maroon">l</font>
;<br/><b>take </b>
<font color="Maroon">r</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_4_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="E1:49_1_1_4_1_1_1"/>
( ( <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; <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> ) or ( <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> &amp; <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> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:49_1_1_4_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/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:49_1_1_4_1_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">G</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:49_1_1_4_1_1"/>
( ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> ) or ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_4_1_2"/><i><font color="Green">E127</font></i>: 
( <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> ) ) )
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>
;<br/><b>take </b>
<font color="Maroon">l</font>
;<br/><b>take </b>
<font color="Maroon">r</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_4_1_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="E1:49_1_1_4_1_2_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> &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/><b>hence </b><a NAME="E2:49_1_1_4_1_2_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/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_4_1_2"/>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">G</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:49_1_1_4_1_2"/>
( ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> ) or ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E15:49_1_1"/><i><font color="Green">E128</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>,<font color="Olive">r</font>) where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>,<font color="Olive">r</font>] </span>}</span>  &amp; ( for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>,<font color="Olive">r</font>) where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>,<font color="Olive">r</font>] </span>}</span>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">A</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E16:49_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A'</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:49_1_1_5"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A'</font> )
 ;<br/>

<b>then 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="E3:49_1_1_5"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">l</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>]
 <b>by </b><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="E5:49_1_1_5"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">A'</font>,<font color="Maroon">l'</font>,<font color="Maroon">r'</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A'</font>]
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_5_1"/>
( ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> ) or ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> ) )
 </span><b>by </b><i><a class="ref" href="chain_1.html#D2" target="_self">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_5_1_1"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> )
 ;<br/><div class="add"><a NAME="E2:49_1_1_5_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l'</font>,<font color="Maroon">r'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:49_1_1_5_1_1"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E4:49_1_1_5_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_5_1_2"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> &amp; <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> )
 ;<br/><div class="add"><a NAME="E2:49_1_1_5_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l'</font>,<font color="Maroon">r'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:49_1_1_5_1_2"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E4:49_1_1_5_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">X</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">G</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<a NAME="E18:49_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">lr</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_1_1_6"/>
<font color="Maroon">lr</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_1_1_6"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">lr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</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="E5:49_1_1_6"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">l</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>]
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:49_1_1_6"/>
( 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="chain_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E7:49_1_1_6"/>
<font color="Maroon">lr</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="chain_1.html#D2" target="_self">Def2</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:49_1_1"/><b>then </b><i><font color="Green">E138</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E20:49_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">l</font>,<font color="Olive">r</font>) where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>,<font color="Olive">r</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font></span>)</span>
 
<b>from </b><i><a class="ref" href="chain_1.html#S1" target="_self">CHAIN_1:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E21:49_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="chain_1.html#K4" target="_self">cell</a> <font color="Olive">l</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">l</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Olive">r</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> : (  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="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> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">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="Olive">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="Olive">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Olive">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="Olive">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="Olive">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="Olive">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="Olive">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>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="frechet2.html#T50">FRECHET2:50</a></i>;<br/>


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