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

<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="E1:86"/><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="E2:86"/><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="E3:86"/><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="E4:86"/><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>
 ;<br/>

<a NAME="E5:86"/><i><font color="Green">E46</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#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a></i>;<br/>
<b>assume </b><a NAME="E6:86"/>
not 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E7:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:86"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:86"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E10:86"/><b>then </b>
<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#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E11:86"/>
contradiction
 <b>by </b><i><a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>


</div>
