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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <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#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> holds <br/> <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <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="Olive" title="b1">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></span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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#K3" title="RCOMP_1:func.3">[.</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#K3" title="RCOMP_1:func.3">.[</a></span> implies  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <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> )</span><br/>



<b>assume </b><a NAME="E1:47"/>
<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#K3" title="RCOMP_1:func.3">[.</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#K3" title="RCOMP_1:func.3">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <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></span><br/>

<a NAME="E2:47"/><b>then </b>
<font color="Maroon" title="c1">A</font> <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"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c2">a</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xxreal_1.html#T224" title="XXREAL_1:th.224">XXREAL_1:224</a>, <a class="ref" href="xxreal_1.html#T294" title="XXREAL_1:th.294">XXREAL_1:294</a></i>;<br/>
<a NAME="E3:47"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c2">a</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T77" title="BORSUK_5:th.77">BORSUK_5:77</a></i>;<br/>
<a NAME="E4:47"/><b>then </b>
<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"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</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"><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>
 
<b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xxreal_1.html#T224" title="XXREAL_1:th.224">XXREAL_1:224</a>, <a class="ref" href="xxreal_1.html#T288" title="XXREAL_1:th.288">XXREAL_1:288</a></i>;<br/>
<b>hence </b><a NAME="E5:47"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <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>
 <b>by </b><i><a class="ref" href="tops_1.html#D1" title="TOPS_1:def.1">TOPS_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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