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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 ;<br/>

<a NAME="E3:50"/><i><font color="Green" title="E44">A3</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:50"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="rcomp_3.html#E2:50"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="borsuk_5.html#T59" title="BORSUK_5:th.59">BORSUK_5:59</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">RO</font> =  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">LC</font> =  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span>;<br/>
<a NAME="E4:50"/><i><font color="Green" title="E45">A4</font></i>: 
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green" title="E46">A5</font></i>: 
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E47">A6</font></i>: 
<span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T19" target="_self" title="RCOMP_3:th.19">Th19</a></i>;<br/>
<i><font color="Green" title="E48">A7</font></i>:  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> = 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E2:50"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="topreal6.html#T80" title="TOPREAL6:th.80">TOPREAL6:80</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E6:50"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="rcomp_3.html#T3" target="_self" title="RCOMP_3:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E5:50"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="topreal6.html#T80" title="TOPREAL6:th.80">TOPREAL6:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:50"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="topreal6.html#T80" title="TOPREAL6:th.80">TOPREAL6:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:50"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="rcomp_3.html#E5:50"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="borsuk_5.html#T75" title="BORSUK_5:th.75">BORSUK_5:75</a></i>
;<br/>
<a NAME="E8:50"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:50"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="rcomp_3.html#T20" target="_self" title="RCOMP_3:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E9:50"/>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:50"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="rcomp_3.html#E7:50"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="tops_1.html#D2" title="TOPS_1:def.2">TOPS_1:def 2</a></i>;<br/>


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