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<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>



<b>assume </b><a NAME="E1:76"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,1</span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/>

<a NAME="E2:76"/><i><font color="Green">E32</font></i>: 
( 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1"/>
(  not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><b>suppose </b></a><a NAME="E1:76_1_1"/>
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:76_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T7">RCOMP_2:7</a></i>;<br/><a NAME="E3:76_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E4:76_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:76_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E6:76_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E6:76_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:76_1_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E3:76_1_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> 
 
;<br/>
<a NAME="E4:76_1_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E6:76_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:76_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:76_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E8:76_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_4.html#K1" target="_self">I(01)</a> </span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<font color="Maroon">a</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E9:76_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_4.html#K1" target="_self">I(01)</a> </span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:76_1_1"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2"><b>suppose </b></a><a NAME="E1:76_1_2"/>
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:76_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_4.html#K1" target="_self">I(01)</a> 
 ;<br/><b>hence </b><a NAME="E3:76_1_2"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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