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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="topmetr.html#K5">I[01]</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="topmetr.html#K3">R^1</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of  <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topgen_5.html#T76" target="_self">Th76</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span> ;<br/>
<a NAME="E2:89"/><i><font color="Green">E68</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:89_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></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="89_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:89_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:89_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:89_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:89_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="topgen_5.html#E5:89_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topgen_5.html#E7:89_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E9:89_1_1"/><i><font color="Green">E72</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E5:89_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topgen_5.html#E7:89_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_1_1"/>
( ( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_1"><b>suppose </b></a><a NAME="E1:89_1_1_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:89_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>11</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topgen_5.html#E5:89_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topgen_5.html#E9:89_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topgen_5.html#T8" target="_self">Th8</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E3:89_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:89_1_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E4:89_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:89_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:89_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_2"><b>suppose </b></a><a NAME="E1:89_1_1_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:89_1_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:89_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreala.html#T10">TOPREALA:10</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E3:89_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E4:89_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_3"><b>suppose </b></a><a NAME="E1:89_1_1_1_3"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:89_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:89_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreala.html#T19">TOPREALA:19</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E3:89_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E4:89_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:89_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topgen_5.html#E1:89_1_1_1_3"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E5:89_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:89_1_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_4"><b>suppose </b></a><a NAME="E1:89_1_1_1_4"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:89_1_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topgen_5.html#E5:89_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topgen_5.html#E9:89_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topgen_5.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E3:89_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:89_1_1_1_4"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E4:89_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:89_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:89_1_1_1_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">c<sub>8</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="E2:89_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:89_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:89_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span>  or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,2</span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span>  or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="topgen_5.html#E4:89_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1"><b>suppose </b></a><a NAME="E1:89_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="topreala.html#T10">TOPREALA:10</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:89_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  )
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/><b>hence </b><a NAME="E4:89_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_2"><b>suppose </b></a><a NAME="E1:89_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:89_1_2_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E5:89_1_2_2"/>
( 0 <a href="hidden.html#R2">in</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> &amp; 0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E6:89_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</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> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topgen_5.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><b>hence </b><a NAME="E7:89_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_3"><b>suppose </b></a><a NAME="E1:89_1_2_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:89_1_2_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,2</span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E5:89_1_2_3"/>
( 1 <a href="hidden.html#R2">in</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> &amp; 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E6:89_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</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> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topgen_5.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><b>hence </b><a NAME="E7:89_1_2_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_4"><b>suppose </b></a><a NAME="E1:89_1_2_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:89_1_2_4"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E5:89_1_2_4"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_2_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:89_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E7:89_1_2_4"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E8:89_1_2_4"/>
<font color="Maroon">c<sub>12</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="topgen_5.html#E3:89_1_2_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreala.html#T19">TOPREALA:19</a></i>;<br/><a NAME="E9:89_1_2_4"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</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> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  )
 <b>by </b><i><a class="txt" href="topgen_5.html#E3:89_1_2_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topgen_5.html#E7:89_1_2_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E10:89_1_2_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:89"/><i><font color="Green">E69</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of  <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="topgen_5.html#T75" target="_self">Th75</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> 3</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>, <font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 3</span>)</span>,1</span><a href="rcomp_2.html#K2">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topreala.html#T11">TOPREALA:11</a>, <a class="ref" href="topreala.html#T15">TOPREALA:15</a></i>;<br/>
<a NAME="E5:89"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span>  )
 
;<br/>
<a NAME="E6:89"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:89"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:89"/>
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="topgen_5.html#T10" target="_self">Th10</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<b>hence </b><a NAME="E9:89"/>
<span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of  <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="topgen_5.html#E2:89"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topgen_5.html#E3:89"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topgen_5.html#E7:89"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topgen_5.html#T77" target="_self">Th77</a></i>;<br/>


</div>
