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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="rcomp_3.html#M1" target="_self">IntervalCover</a> of <font color="Maroon">F</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="rcomp_3.html#V1" target="_self">connected</a>
 <b>and </b><br/><a NAME="E4:87"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:87"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E6:87"/><i><font color="Green">E47</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E7:87"/>
not <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E8:87"/><b>then </b>
not  <a href="relset_1.html#K5">rng</a> <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E9:87"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/>
<a NAME="E10:87"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:87"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:87_1"/>
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> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> 1 holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a>, <a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:87_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:87_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:87_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:87_1"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:87_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">1</span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:87_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="rcomp_3.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:87"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div>
