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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="rcomp_3.html#M1" target="_self">IntervalCover</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="rcomp_3.html#M2" target="_self">IntervalCoverPts</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:100"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:100"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_2.html#V1">open</a>
 <b>and </b><br/><a NAME="E3:100"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="rcomp_3.html#V1" target="_self">connected</a>
 <b>and </b><br/><a NAME="E4:100"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/>
<b>assume </b><b>that </b><br/><a NAME="E5:100"/><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:100"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E7:100"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>3</sub></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<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E6:100"><i><font color="Green">E76</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="rcomp_3.html#E5:100"><i><font color="Green">E75</font></i></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="100_1_1"><b>suppose </b></a><a NAME="E1:100_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:100_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:100"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rcomp_3.html#E2:100"><i><font color="Green">E72</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rcomp_3.html#D3" target="_self">Def3</a>, <a class="txt" href="rcomp_3.html#E3:100"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E3:100_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:100"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rcomp_3.html#E2:100"><i><font color="Green">E72</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rcomp_3.html#D2" target="_self">Def2</a>, <a class="txt" href="rcomp_3.html#E3:100"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E4:100_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></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="E5:100_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:100"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rcomp_3.html#E2:100"><i><font color="Green">E72</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100"><i><font color="Green">E74</font></i></a>, <a class="txt" href="rcomp_3.html#E7:100"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rcomp_3.html#D2" target="_self">Def2</a>, <a class="txt" href="rcomp_3.html#E3:100"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E6:100_1_1"/>
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<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E6:100"><i><font color="Green">E76</font></i></a>, <a class="txt" href="rcomp_3.html#E1:100_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:100_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:100_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:100_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:100_1_1"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E10:100_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E7:100_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:100_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:100_1_1"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E12:100_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:100"><i><font color="Green">E74</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E13:100_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T3">RCOMP_1:3</a></i>;<br/><a NAME="E14:100_1_1"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E5:100_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="rcomp_3.html#E12:100_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rcomp_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E15:100_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></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<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E2:100_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="rcomp_3.html#E1:100_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_2"><b>suppose </b></a><a NAME="E1:100_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:100_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:100_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:100_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="rcomp_3.html#E2:100_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:100_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T17">XREAL_1:17</a></i>;<br/><a NAME="E6:100_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:100_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:100_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></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<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:100"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rcomp_3.html#E2:100"><i><font color="Green">E72</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rcomp_3.html#D3" target="_self">Def3</a>, <a class="txt" href="rcomp_3.html#E3:100"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E8:100_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="rcomp_3.html#E6:100"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:100_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</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">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E6:100_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:100_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:100"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rcomp_3.html#E2:100"><i><font color="Green">E72</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100"><i><font color="Green">E74</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="rcomp_3.html#T76" target="_self">Th76</a>, <a class="txt" href="rcomp_3.html#E3:100"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E11:100_1_2"/>
<font color="Maroon">c<sub>3</sub></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<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E6:100"><i><font color="Green">E76</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:100_1_2"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></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<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:100"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rcomp_3.html#E2:100"><i><font color="Green">E72</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100"><i><font color="Green">E74</font></i></a>, <a class="txt" href="rcomp_3.html#E4:100_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="rcomp_3.html#E3:100_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="rcomp_3.html#D2" target="_self">Def2</a>, <a class="txt" href="rcomp_3.html#E3:100"><i><font color="Green">E73</font></i></a></i>;<br/><b>hence </b><a NAME="E13:100_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></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<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E10:100_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="rcomp_3.html#E3:100_1_2"><i><font color="Green">E79</font></i></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>
