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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:87"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:87"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E4:87"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E5:87"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:87"><i><font color="Green">E75</font></i></a>, <a class="ref" href="rcomp_1.html#T31">RCOMP_1:31</a></i>;<br/>
<a NAME="E6:87"/><i><font color="Green">E78</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:87"><i><font color="Green">E75</font></i></a>, <a class="txt" href="borsuk_4.html#E4:87"><i><font color="Green">E76</font></i></a>, <a class="ref" href="rcomp_1.html#T30">RCOMP_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E8:87"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:87_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:87_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E4:87_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E1:87_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="borsuk_4.html#T51" target="_self">Th51</a></i>;<br/>
<b>hence </b><a NAME="E5:87_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E10:87"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E4:87"><i><font color="Green">E76</font></i></a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E11:87"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<a NAME="E12:87"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="connsp_1.html#V2">connected</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E5:87"><i><font color="Green">E77</font></i></a>, <a class="txt" href="borsuk_4.html#E6:87"><i><font color="Green">E78</font></i></a>, <a class="ref" href="borsuk_4.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E13:87"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E14:87"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E15:87"/><i><font color="Green">E84</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E14:87"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_4.html#E11:87"><i><font color="Green">E80</font></i></a>, <a class="txt" href="borsuk_4.html#E85"><i><font color="Green">Lemma72</font></i></a></i>;<br/>
<a NAME="E16:87"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E12:87"><i><font color="Green">E81</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E17:87"/><i><font color="Green">E86</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_2"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:87_2"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E13:87"><i><font color="Green">E82</font></i></a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E3:87_2"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:87"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="txt" href="borsuk_4.html#E5:87"><i><font color="Green">E77</font></i></a>, <a class="txt" href="borsuk_4.html#E15:87"><i><font color="Green">E84</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:87"/><i><font color="Green">E87</font></i>: 
not 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_3"/>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:87_3"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E13:87"><i><font color="Green">E82</font></i></a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E3:87_3"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:87"><i><font color="Green">E78</font></i></a>, <a class="txt" href="borsuk_4.html#E15:87"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_4.html#E1:87"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:87"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:87_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>thus </b><a NAME="E3:87_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="borsuk_4.html#T10" target="_self">Th10</a>, <a class="txt" href="borsuk_4.html#E16:87"><i><font color="Green">E85</font></i></a>, <a class="txt" href="borsuk_4.html#E1:87_4"><i><font color="Green">E89</font></i></a>, <a class="txt" href="borsuk_4.html#E17:87"><i><font color="Green">E86</font></i></a>, <a class="txt" href="borsuk_4.html#E18:87"><i><font color="Green">E87</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E11:87"><i><font color="Green">E80</font></i></a>, <a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E21:87"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E8:87"><i><font color="Green">E79</font></i></a>, <a class="txt" href="borsuk_4.html#E15:87"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_4.html#E19:87"><i><font color="Green">E88</font></i></a></i>;<br/>


</div>
