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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:100"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/>

<b>thus </b><a NAME="E2:100"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E3:100_1"/><i><font color="Green">E80</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_6.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E4:100_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:100_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:100_1"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:100_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E6:100_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E7:100_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:100"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E3:100"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E3:100_2"/><i><font color="Green">E80</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_6.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:100_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:100_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:100_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:100_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:100_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E7:100_2"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:100"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:100"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 ;<br/>


</div>
