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

<b>set </b><font color="Maroon">KK</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Olive">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:87"/>
<a href="borsuk_6.html#NK13" target="_self">ICC</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Olive">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><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">x</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"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 ;<br/>

<a NAME="E2:87_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E4:87_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E8:87"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E6:87_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="mcart_1.html#K2">`2</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="mcart_1.html#K2">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:87_1"/>
<font color="Maroon">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T15" target="_self">Th15</a>, <a class="ref" href="borsuk_6.html#T16" target="_self">Th16</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E8:87_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Olive">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T15" target="_self">Th15</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:87"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Olive">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E4:87"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 ;<br/>
<a NAME="E5:87"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:87"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E7:87"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T15" target="_self">Th15</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E8:87"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a>  <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>  &amp; <font color="Maroon">p</font> <a href="mcart_1.html#K2">`2</a>  <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>  )
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T15" target="_self">Th15</a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E9:87"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T16" target="_self">Th16</a>, <a class="ref" href="borsuk_6.html#T17" target="_self">Th17</a>, </i>;<br/>
<a NAME="E10:87"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T17" target="_self">Th17</a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E11:87"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E12:87"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></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/>
<a NAME="E13:87"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; 0 <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> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T221">XREAL_1:221</a>, <a class="ref" href="xreal_1.html#T222">XREAL_1:222</a></i>;<br/>
<a NAME="E14:87"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T16" target="_self">Th16</a>, <a class="ref" href="borsuk_6.html#T17" target="_self">Th17</a>, </i>;<br/>
<b>hence </b><a NAME="E15:87"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
