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<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<b>assume </b><a NAME="E2:80"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default"><font color="Maroon">a</font></span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 ;<br/>

<a NAME="E3:80"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">A'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E14"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">a</font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:80"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a>, </i>;<br/>
<a NAME="E7:80"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E8:80"/>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a>, </i>;<br/>
<a NAME="E9:80"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T18" target="_self">Th18</a>, <a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> or  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> )
 </span><b>by </b><i><a class="ref" href="borsuk_5.html#T19" target="_self">Th19</a>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><b>suppose </b></a><a NAME="E1:80_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:80_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">a</font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2"><b>suppose </b></a><a NAME="E1:80_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 ;<br/><div class="add"><a NAME="E2:80_1_2"/><b>then </b>
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="borsuk_5.html#T17" target="_self">Th17</a>, <a class="ref" href="borsuk_5.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E3:80_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">a</font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a>, <a class="ref" href="borsuk_5.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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