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

<a NAME="E1:101_1_1"/><i><font color="Green">E73</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E2:101_1_1"/><i><font color="Green">E74</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> :  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:101_1_1"/><i><font color="Green">E75</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E4:101_1_1"/><i><font color="Green">E76</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> :  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E5:101_1_1"/><i><font color="Green">E77</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:101_1_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:101_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>thus </b><a NAME="E2:101_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:101_1_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:101_1_1_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:101_1_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_1"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:101_1_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E8:101_1_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E9:101_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:101_1_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:101_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:101_1_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_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="E1:101_1_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"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:101_1_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:101_1_1_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>
<a NAME="E5:101_1_1_1_1_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:101_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:101_1_1_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="borsuk_1.html#T75" target="_self">Th75</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="borsuk_1.html#E3:101_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="borsuk_1.html#E10:101_1_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:101_1_1_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_1_1_1_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:101_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E8:101_1_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="borsuk_1.html#E1:101_1_1_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:101_1_1_1_1_2_1_1_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:101_1_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>thus </b><a NAME="E5:101_1_1_1_1_2_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_1_2_1_1_1_1"><i><font color="Green">E84</font></i></a>, <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="101_1_1_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:101_1_1_1_1_2_1_1_1_2"/><i><font color="Green">E84</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/><b>thus </b><a NAME="E2:101_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:101_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:101_1_1_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_1.html#E1:101_1_1_1_1_2_1_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:101_1_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:101_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:101_1_1_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:101_1_1_1_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1_1_1_2_2_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="borsuk_1.html#E2:101_1_1_1_1_2"><i><font color="Green">E83</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="101_1_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:101_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:101_1_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="borsuk_1.html#E2:101_1_1_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E3:101_1_1_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E8:101_1_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:101_1_1_1_1_2_2_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:101_1_1_1_1_2_2_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:101_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:101_1_1_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_1.html#E3:101_1_1_1_1_2_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:101_1_1_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_1_2_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:101_1_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E14:101_1_1_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E15:101_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:101_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="borsuk_1.html#E4:101_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="borsuk_1.html#E7:101_1_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="borsuk_1.html#E14:101_1_1_1_1"><i><font color="Green">E83</font></i></a></i>;<br/>

<a NAME="E16:101_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E17:101_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>9</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="E18:101_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E19:101_1_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_1_3"/>
<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="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:101_1_1_1_1_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:101_1_1_1_1_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:101_1_1_1_1_3"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_1_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:101_1_1_1_1_3"/><b>then </b>
not <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T79" target="_self">Th79</a></i>;<br/><b>hence </b><a NAME="E7:101_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_1.html#E19:101_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_1.html#E3:101_1_1_1_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="borsuk_1.html#E5:101_1_1_1_1_3"><i><font color="Green">E87</font></i></a>, <a class="ref" href="borsuk_1.html#T77" target="_self">Th77</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:101_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E19:101_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span><b> such that </b><a NAME="E3:101_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:101_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:101_1_1_1"/><i><font color="Green">E80</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="borsuk_1.html#E3:101_1_1_1"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E6:101_1_1_1"/><i><font color="Green">E81</font></i>: 
(  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:101_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_1.html#E3:101_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="borsuk_1.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:101_1_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_1_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:101_1_1_1_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> )
 
<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">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:101_1_1_1_2_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E5:101_1_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_2_1"><i><font color="Green">E83</font></i></a></i>;<br/>

<a NAME="E6:101_1_1_1_2_1"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E2:101_1_1_1_2_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="borsuk_1.html#T80" target="_self">Th80</a></i>;<br/>
<a NAME="E7:101_1_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E8:101_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E9:101_1_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="borsuk_1.html#E4:101_1_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_1.html#E6:101_1_1_1_2_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:101_1_1_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E3:101_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="borsuk_1.html#E2:101_1_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:101_1_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:101_1_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:101_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:101_1_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E6:101_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E6:101_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="borsuk_1.html#E4:101_1_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E8:101_1_1_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E9:101_1_1_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:101_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>hence </b><a NAME="E10:101_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:101_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_1.html#E2:101_1_1"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="borsuk_1.html#K14" target="_self">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E8:101_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="borsuk_1.html#V1" target="_self">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D14" target="_self">BORSUK_1:def 14</a><br/>

<b>assume </b><a NAME="E1:101_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="borsuk_1.html#E3:101_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="borsuk_1.html#T25" target="_self">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E4:101_1_1_2"/><i><font color="Green">E79</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E3:101_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E5:101_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:101_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_1.html#E1:101_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E6:101_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E7:101_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E8:101_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:101_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:101_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="borsuk_1.html#T69" target="_self">Th69</a></i>;<br/>
<a NAME="E10:101_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:101_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:101_1_1"/>
 <a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1" target="_self">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="borsuk_1.html#K14" target="_self">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


</div>
