<?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>let </b><font color="Maroon">G</font> be    <a href="rcomp_3.html#M2" target="_self">IntervalCoverPts</a> of <font color="Maroon">C</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:101"/><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="E2:101"/><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="E3:101"/><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="E4:101"/><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/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E5:101"/><i><font color="Green">E46</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">G</font>
 <b>and </b><br/><a NAME="E6:101"/><i><font color="Green">E47</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">G</font>
 <b>and </b><br/><a NAME="E7:101"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <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">G</font> implies <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E8:101"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E9:101"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</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="101_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:101_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="E2:101_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>and </b><br/><a NAME="E3:101_1"/><i><font color="Green">E53</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">G</font>
 <b>and </b><br/><a NAME="E4:101_1"/><i><font color="Green">E91</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">G</font>
 ;<br/>

<a NAME="E5:101_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">m</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#T36" target="_self">Th36</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:101_1"/><i><font color="Green">E96</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T37" target="_self">Th37</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:101_1"/><i><font color="Green">E97</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#T39" target="_self">Th39</a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:101_1"/><i><font color="Green">E98</font></i>: 
<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">G</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:101_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:101_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:101_1"/><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <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
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <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">G</font> ) or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:101_1_1_1"/><i><font color="Green">E101</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E2:101_1_1_1"/><i><font color="Green">E102</font></i>: 
<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">G</font>
 ;<br/><div class="add"><a NAME="E3:101_1_1_1"/>
<font color="Maroon">k</font> <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#T43" target="_self">Th43</a>, <a class="ref" href="rcomp_3.html#T42" target="_self">Th42</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:101_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <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#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="txt" href="rcomp_3.html#E4:101_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E5:101_1_1_1"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="txt" href="rcomp_3.html#E4:101_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T43" target="_self">Th43</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a></i>;<br/><a NAME="E6:101_1_1_1"/><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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E7:101_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:101_1_1_1"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T39" target="_self">Th39</a>, <a class="ref" href="rcomp_3.html#T35" target="_self">Th35</a>, <a class="ref" href="rcomp_3.html#T37" target="_self">Th37</a>, <a class="txt" href="rcomp_3.html#E4:101_1"><i><font color="Green">E91</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:101_1_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="rcomp_3.html#T36" target="_self">Th36</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:101_1_1_2"/><i><font color="Green">E106</font></i>: 
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="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> or 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:101_1_1_2_1_1"/><i><font color="Green">E107</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_2_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:101_1_1_2_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
 <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> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> 2
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E4:101_1_1_2_1_1"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> 1 <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">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, , <a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E5:101_1_1_2_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> 2
 <b>by </b><i><a class="txt" href="rcomp_3.html#E5:101_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:101_1_1_2_1_1"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:101_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="rcomp_3.html#T43" target="_self">Th43</a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_2_1_2"/>
1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_2_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E3:101_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E4:101_1_1_2_1_2"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:101_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="rcomp_3.html#T43" target="_self">Th43</a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_3"><b>suppose </b></a><a NAME="E1:101_1_1_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:101_1_1_3_1_1"/><i><font color="Green">E111</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">z</font> = <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E2:101_1_1_3_1_1"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:101_1_1_3_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:101_1_1_3_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:101_1_1_3_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:101_1_1_3_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E7:101_1_1_3_1_1"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T36" target="_self">Th36</a>, <a class="txt" href="rcomp_3.html#E4:101_1"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E8:101_1_1_3_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</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">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, , <a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="txt" href="rcomp_3.html#E5:101_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E9:101_1_1_3_1_1"/><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E10:101_1_1_3_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</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#T42" target="_self">Th42</a>, <a class="txt" href="rcomp_3.html#E7:101_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:101_1_1_3_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 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="txt" href="rcomp_3.html#E6:101_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:101_1_1_3_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <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="E13:101_1_1_3_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E10:101_1"><i><font color="Green">E99</font></i></a></i>;<br/><a NAME="E14:101_1_1_3_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E15:101_1_1_3_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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/><a NAME="E16:101_1_1_3_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</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#T29" target="_self">Th29</a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E17:101_1_1_3_1_1"/><i><font color="Green">E121</font></i>: 
not <font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="txt" href="rcomp_3.html#E6:101_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="rcomp_3.html#E11:101_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E18:101_1_1_3_1_1"/>
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T4">RCOMP_1:4</a></i>;<br/><a NAME="E19:101_1_1_3_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</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">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <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>, , </i>;<br/><a NAME="E20:101_1_1_3_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</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">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E21:101_1_1_3_1_1"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E8:101_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="rcomp_3.html#T43" target="_self">Th43</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_3_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_1_1_3_1_2"/>
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</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#T43" target="_self">Th43</a>, <a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:101"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</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#T33" target="_self">Th33</a>, <a class="ref" href="rcomp_3.html#T34" target="_self">Th34</a>);<br/></i>
<b>hence </b><a NAME="E11:101"/>
not <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a></i>;<br/>


</div>
