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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">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</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="E1:88"/><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="E2:88"/><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="E3:88"/><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="E4:88"/><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="E5:88"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E6:88"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E7:88"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></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">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E8:88"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E9:88"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:88_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E2:88_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:88_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:88_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1"><b>suppose </b></a><a NAME="E1:88_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:88_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:88_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:88_1_1_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</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_3.html#T26">FINSEQ_3:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_2"><b>suppose </b></a><a NAME="E1:88_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:88_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:88_1_1_2"/><i><font color="Green">E96</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:88_1_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:88_1_1_2"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</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#T30" target="_self">Th30</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/><b>hence </b><a NAME="E6:88_1_1_2"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T25" target="_self">Th25</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:88"/>
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>);<br/></i>
<b>hence </b><a NAME="E11:88"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</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></i>;<br/>


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