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

<a NAME="E1:100_1_1"/><i><font color="Green" title="E72">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<a NAME="E2:100_1_1"/><i><font color="Green" title="E73">A2</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">D</font> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<a NAME="E3:100_1_1"/><i><font color="Green" title="E74">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<a NAME="E4:100_1_1"/><i><font color="Green" title="E75">A4</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<a NAME="E5:100_1_1"/><i><font color="Green" title="E76">A5</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:100_1_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:100_1_1"><i><font color="Green" title="E76">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span>;<br/><b>thus </b><a NAME="E2:100_1_1_1"/>
( <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span> implies  ex <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_1_1_1_1"/>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">D</font><b> such that </b><br/><a NAME="E3:100_1_1_1_1"/><i><font color="Green" title="E77">A6</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1"/><i><font color="Green" title="E78">A7</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:100_1_1"><i><font color="Green" title="E73">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_1"><i><font color="Green" title="E78">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:100_1_1_1_1"/><i><font color="Green" title="E79">A8</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E8:100_1_1_1_1"/><i><font color="Green" title="E80">A9</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_1"><i><font color="Green" title="E78">A7</font></i></a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E9:100_1_1_1_1"/>
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:100_1_1_1_1"/><b>then </b><i><font color="Green" title="E81">A10</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:100_1_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:100_1_1_1_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:100_1_1_1_1_1"/><i><font color="Green" title="E82">A11</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1_1"/><i><font color="Green" title="E83">A12</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ;<br/>
<a NAME="E5:100_1_1_1_1_1"/>
not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_1_1"><i><font color="Green" title="E83">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:100_1_1_1_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1_1_1_1"><i><font color="Green" title="E82">A11</font></i></a>, <a class="ref" href="borsuk_1.html#T75" target="_self" title="BORSUK_1:th.75">Th75</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">Q</font> = <font color="Maroon" title="c4">P</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E10:100_1_1_1_1"><i><font color="Green" title="E81">A10</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:100_1_1_1_1_2"/>
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> implies  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_1_1_1_1_2_1"/><i><font color="Green" title="E82">A13</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">e</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_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="E1:100_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> or  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:100_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font>
 ;<br/><div class="add"><a NAME="E2:100_1_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E8:100_1_1_1_1"><i><font color="Green" title="E80">A9</font></i></a>, <a class="txt" href="borsuk_1.html#E1:100_1_1_1_1_2_1"><i><font color="Green" title="E82">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:100_1_1_1_1_2_1_1_1_1"/><i><font color="Green" title="E83">A14</font></i>: 
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">u</font> &amp; <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1_1_1"><i><font color="Green" title="E77">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>take </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font>;<br/><b>thus </b><a NAME="E5:100_1_1_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">u</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_1_2_1_1_1_1"><i><font color="Green" title="E83">A14</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:100_1_1_1_1_2_1_1_1_2"/><i><font color="Green" title="E83">A15</font></i>: 
not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">u</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>thus </b><a NAME="E2:100_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">u</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:100_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1_1_1_2_1"><i><font color="Green" title="E82">A13</font></i></a>, <a class="txt" href="borsuk_1.html#E1:100_1_1_1_1_2_1_1_1_2"><i><font color="Green" title="E83">A15</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:100_1_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:100_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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="E4:100_1_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> )
 ;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c10">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:100_1_1_1_1_2"/><i><font color="Green" title="E82">A16</font></i>: 
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">u</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_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="E1:100_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1_1_1_2"><i><font color="Green" title="E82">A16</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 ;<br/><div class="add"><a NAME="E2:100_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1_1_1"><i><font color="Green" title="E77">A6</font></i></a>, <a class="txt" href="borsuk_1.html#E3:100_1_1_1_1_2"><i><font color="Green" title="E82">A16</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E3:100_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E8:100_1_1_1_1"><i><font color="Green" title="E80">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:100_1_1_1_1_2_2_1_2"/><i><font color="Green" title="E83">A17</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1_2_2_1_2"/><i><font color="Green" title="E84">A18</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ;<br/><a NAME="E5:100_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1_1_1_2"><i><font color="Green" title="E82">A16</font></i></a>, <a class="txt" href="borsuk_1.html#E3:100_1_1_1_1_2_2_1_2"><i><font color="Green" title="E83">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:100_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_1_2_2_1_2"><i><font color="Green" title="E84">A18</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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="E5:100_1_1_1_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E14:100_1_1_1_1"/><b>then </b><i><font color="Green" title="E82">A19</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Q</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">Q</font>
;<br/>

<b>thus </b><a NAME="E15:100_1_1_1_1"/>
<font color="Maroon" title="c8">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E4:100_1_1"><i><font color="Green" title="E75">A4</font></i></a>, <a class="txt" href="borsuk_1.html#E7:100_1_1_1_1"><i><font color="Green" title="E79">A8</font></i></a>, <a class="txt" href="borsuk_1.html#E14:100_1_1_1_1"><i><font color="Green" title="E82">A19</font></i></a></i>;<br/>

<a NAME="E16:100_1_1_1_1"/>
( <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E17:100_1_1_1_1"/>
<font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E18:100_1_1_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E19:100_1_1_1_1"/><b>then </b><i><font color="Green" title="E83">A20</font></i>: 
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> &amp; <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_1_1_1_1_3"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c10">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:100_1_1_1_1_3"/><i><font color="Green" title="E84">A21</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1_3"/><i><font color="Green" title="E85">A22</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ;<br/><a NAME="E5:100_1_1_1_1_3"/><i><font color="Green" title="E86">A23</font></i>: 
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_1_3"><i><font color="Green" title="E85">A22</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:100_1_1_1_1_3"/><b>then </b>
not <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T79" target="_self" title="BORSUK_1:th.79">Th79</a></i>;<br/><b>hence </b><a NAME="E7:100_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_1.html#E19:100_1_1_1_1"><i><font color="Green" title="E83">A20</font></i></a>, <a class="txt" href="borsuk_1.html#E3:100_1_1_1_1_3"><i><font color="Green" title="E84">A21</font></i></a>, <a class="txt" href="borsuk_1.html#E5:100_1_1_1_1_3"><i><font color="Green" title="E86">A23</font></i></a>, <a class="ref" href="borsuk_1.html#T77" target="_self" title="BORSUK_1:th.77">Th77</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:100_1_1_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E19:100_1_1_1_1"><i><font color="Green" title="E83">A20</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c5">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span><b> such that </b><a NAME="E4:100_1_1_1"/><i><font color="Green" title="E77">A24</font></i>: 
<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:100_1_1_1"/><i><font color="Green" title="E78">A25</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 ;<br/><a NAME="E6:100_1_1_1"/><i><font color="Green" title="E79">A26</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> st <br/>( <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1"><i><font color="Green" title="E75">A4</font></i></a>, <a class="txt" href="borsuk_1.html#E4:100_1_1_1"><i><font color="Green" title="E77">A24</font></i></a></i>;<br/><a NAME="E7:100_1_1_1"/><i><font color="Green" title="E80">A27</font></i>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Q</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">P</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E3:100_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T25" target="_self" title="BORSUK_1:th.25">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:100_1_1_1_2"/>
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span> implies  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:100_1_1_1_2_1"/><i><font color="Green" title="E81">A28</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ;<br/>

<a NAME="E2:100_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:100_1_1_1_2_1"/><i><font color="Green" title="E82">A29</font></i>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">u</font>
;<br/>

<b>thus </b><a NAME="E5:100_1_1_1_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_2_1"><i><font color="Green" title="E82">A29</font></i></a></i>;<br/>

<a NAME="E6:100_1_1_1_2_1"/><i><font color="Green" title="E83">A30</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1_1_2_1"><i><font color="Green" title="E81">A28</font></i></a>, <a class="ref" href="borsuk_1.html#T80" target="_self" title="BORSUK_1:th.80">Th80</a></i>;<br/>
<a NAME="E7:100_1_1_1_2_1"/>
<font color="Maroon" title="c7">u</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_2_1"><i><font color="Green" title="E82">A29</font></i></a>, <a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<a NAME="E8:100_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_1_2_1"><i><font color="Green" title="E82">A29</font></i></a>, <a class="ref" href="borsuk_1.html#T29" target="_self" title="BORSUK_1:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E9:100_1_1_1_2_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:100_1_1_1"><i><font color="Green" title="E78">A25</font></i></a>, <a class="txt" href="borsuk_1.html#E4:100_1_1_1_2_1"><i><font color="Green" title="E82">A29</font></i></a>, <a class="txt" href="borsuk_1.html#E6:100_1_1_1_2_1"><i><font color="Green" title="E83">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c7">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:100_1_1_1_2"/><i><font color="Green" title="E81">A31</font></i>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> &amp; <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 ;<br/><a NAME="E4:100_1_1_1_2"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:100_1_1_1"><i><font color="Green" title="E78">A25</font></i></a>, <a class="txt" href="borsuk_1.html#E3:100_1_1_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:100_1_1_1_2"/><b>then </b><i><font color="Green" title="E82">A32</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E6:100_1_1_1_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E7:100_1_1_1_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E7:100_1_1_1"><i><font color="Green" title="E80">A27</font></i></a>, <a class="txt" href="borsuk_1.html#E5:100_1_1_1_2"><i><font color="Green" title="E82">A32</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E9:100_1_1_1"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E10:100_1_1_1"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E6:100_1_1_1"><i><font color="Green" title="E79">A26</font></i></a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/><b>hence </b><a NAME="E11:100_1_1_1"/>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E2:100_1_1"><i><font color="Green" title="E73">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">T</font> =  <a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font> as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E8:100_1_1"/>
<font color="Maroon" title="c4">T</font> is <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D14" target="_self" title="BORSUK_1:def.14">BORSUK_1:def 14</a><br/>

<b>assume </b><a NAME="E1:100_1_1_2"/><i><font color="Green" title="E77">A33</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">B</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T25" target="_self" title="BORSUK_1:th.25">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<a NAME="E4:100_1_1_2"/><i><font color="Green" title="E78">A34</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E3:100_1_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T27" target="_self" title="BORSUK_1:th.27">Th27</a></i>;<br/>
<a NAME="E5:100_1_1_2"/>
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:100_1_1"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E1:100_1_1_2"><i><font color="Green" title="E77">A33</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E6:100_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D14" target="_self" title="BORSUK_1:def.14">Def14</a></i>;<br/>
<a NAME="E7:100_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>
<a NAME="E8:100_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:100_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:100_1_1_2"><i><font color="Green" title="E78">A34</font></i></a>, <a class="ref" href="borsuk_1.html#T69" target="_self" title="BORSUK_1:th.69">Th69</a></i>;<br/>
<a NAME="E10:100_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:100_1_1_2"/>
<font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:100_1_1"/>
 <a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c3">D</font> is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="borsuk_1.html#K14" title="BORSUK_1:func.14">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span>
 ;<br/>


</div>
