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

<a NAME="E1:101_1_1"/><i><font color="Green">E24</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">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:101_1_1"/><i><font color="Green">E29</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">D</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> :  <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:101_1_1"/><i><font color="Green">E31</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">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:101_1_1"/><i><font color="Green">E32</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">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> 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">D</font></span>)</span> :  <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:101_1_1"/><i><font color="Green">E33</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">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</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">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font> )
 
<b>by </b><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">D</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">D</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>let </b><font color="Maroon">P</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">D</font></span>)</span>;<br/><b>thus </b><a NAME="E2:101_1_1_1"/>
( <font color="Maroon">P</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">D</font></span>)</span> implies  ex <font color="Olive">Q</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">D</font></span>)</span></span>)</span> st <br/>( <font color="Olive">Q</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">D</font></span>)</span></span>)</span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</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">D</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">P</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">D</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:101_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:101_1_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a></i>;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:101_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:101_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</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">D</font> <a href="tarski.html#R1">c=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</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">E39</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T27" target="_self">Th27</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">x</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</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">D</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">e</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">e</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">x</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:101_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>
<a NAME="E5:101_1_1_1_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T33" target="_self">Th33</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">e</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">P</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">x</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</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">D</font></span>)</span></span>)</span> <b>by </b><i>, , <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">e</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">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies  ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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">E42</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">e</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</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">e</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> or  not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</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">e</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T32" target="_self">Th32</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">u</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">E43</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>thus </b><a NAME="E5:101_1_1_1_1_2_1_1_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T33" target="_self">Th33</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">E44</font></i>: 
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">u</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</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">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T33" target="_self">Th33</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">u</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">y</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</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">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">u</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">E45</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">u</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">x</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> </span>}</span>  )
 </span><b>by </b><i>, <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">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</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">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T32" target="_self">Th32</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">u</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">x</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:101_1_1_1_1_2_2_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/><a NAME="E5:101_1_1_1_1_2_2_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T33" target="_self">Th33</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">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <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">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E14:101_1_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">Q</font>
;<br/>

<b>thus </b><a NAME="E15:101_1_1_1_1"/>
<font color="Maroon">Q</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">D</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#D1" target="_self">Def1</a>, <a class="ref" href="borsuk_1.html#T31" target="_self">Th31</a>, </i>;<br/>

<a NAME="E16:101_1_1_1_1"/>
( <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</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">D</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">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</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">D</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">e</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">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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">D</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E19:101_1_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">e</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">D</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">e</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">x</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:101_1_1_1_1_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/><a NAME="E5:101_1_1_1_1_3"/><i><font color="Green">E107</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T35" target="_self">Th35</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">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:101_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="borsuk_1.html#T33" target="_self">Th33</a>, , <a class="ref" href="borsuk_1.html#T36" target="_self">Th36</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:101_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T33" target="_self">Th33</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">Q</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">D</font></span>)</span></span>)</span><b> such that </b><a NAME="E3:101_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">Q</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">D</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:101_1_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</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">D</font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:101_1_1_1"/><i><font color="Green">E110</font></i>: 
 ex <font color="Olive">A</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">D</font></span>)</span> st <br/>( <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp;  <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#D1" target="_self">Def1</a>, <a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E6:101_1_1_1"/><i><font color="Green">E111</font></i>: 
(  <a href="tarski.html#K3">union</a> <font color="Maroon">Q</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T27" target="_self">Th27</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">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:101_1_1_1_2"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">Q</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">Y</font></span>)</span> implies  ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</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">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">Q</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">Y</font></span>)</span>
 ;<br/>

<a NAME="E2:101_1_1_1_2_1"/><b>then </b><i><font color="Green">E112</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</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">u</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">E147</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>thus </b><a NAME="E5:101_1_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>

<a NAME="E6:101_1_1_1_2_1"/><i><font color="Green">E148</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">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</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">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E7:101_1_1_1_2_1"/>
<font color="Maroon">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:101_1_1_1_2_1"/><b>then </b>
<font color="Maroon">u</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">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:101_1_1_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a>, , <a class="ref" href="borsuk_1.html#T33" target="_self">Th33</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">u</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">E149</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E3:101_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</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">E150</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:101_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">P</font>
 <b>by </b><i>, <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">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">Q</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">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T32" target="_self">Th32</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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">Q</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">Y</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">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T31" target="_self">Th31</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">P</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">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T</font> =  <a href="borsuk_1.html#K14" target="_self">space</a> <font color="Maroon">D</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">D</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">T</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">A</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">D</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">E151</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">B</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</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">D</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:101_1_1_2"/><i><font color="Green">E152</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:101_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a>, <a class="ref" href="borsuk_1.html#T29" target="_self">Th29</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">B</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E7:101_1_1_2"/><b>then </b>
<font color="Maroon">B</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">B</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">X</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">A</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">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a>, </i>;<br/>
<a NAME="E10:101_1_1_2"/><b>then </b>
<font color="Maroon">A</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">A</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">D</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">D</font></span>)</span>
 ;<br/>


</div>
