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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c3">A'</font> = <font color="Maroon" title="c1">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<b>assume </b><a NAME="E2:80"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<a NAME="E3:80"/><b>then </b><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c3">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T63" target="_self" title="BORSUK_5:th.63">Th63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">C</font> = <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c4">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T66" target="_self" title="BORSUK_5:th.66">Th66</a></i>;<br/>
<a NAME="E6:80"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E2:80"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T67" target="_self" title="BORSUK_5:th.67">Th67</a></i>;<br/>
<a NAME="E7:80"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c3">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A'</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E8:80"/>
<font color="Maroon" title="c3">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E2:80"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_1.html#T45" title="XXREAL_1:th.45">XXREAL_1:45</a></i>;<br/>
<a NAME="E9:80"/><b>then </b><i><font color="Green" title="E59">A6</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E5:80"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="tops_1.html#T31" title="TOPS_1:th.31">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#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> or  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A'</font> )
 </span><b>by </b><i><a class="txt" href="borsuk_5.html#E6:80"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="borsuk_5.html#E7:80"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="borsuk_5.html#E9:80"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="borsuk_5.html#T2" target="_self" title="BORSUK_5:th.2">Th2</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#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:80_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</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#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A'</font>
 ;<br/><div class="add"><a NAME="E2:80_1_2"/><b>then </b>
( <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="txt" href="borsuk_5.html#E3:80"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="borsuk_5.html#T57" target="_self" title="BORSUK_5:th.57">Th57</a></i>;<br/><b>hence </b><a NAME="E3:80_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E2:80"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T71" target="_self" title="BORSUK_5:th.71">Th71</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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