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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</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="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">d</font> &amp; <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>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <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"><font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> holds <br/><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</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="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">d</font> &amp; <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="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <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"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> implies <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">d</font> )
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E6">A2</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#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c2">B</font> <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"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 ; <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>,<font color="Maroon" title="c2">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><br/>

<a NAME="E4:14"/>
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> <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="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="borsuk_4.html#T21" title="BORSUK_4:th.21">BORSUK_4:21</a></i>;<br/>
<a NAME="E5:14"/><b>then </b>
 <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_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="topreal6.html#T80" title="TOPREAL6:th.80">TOPREAL6:80</a></i>;<br/>
<a NAME="E6:14"/><b>then </b><i><font color="Green" title="E8">A4</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E3:14"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xxreal_1.html#T90" title="XXREAL_1:th.90">XXREAL_1:90</a></i>;<br/>
<a NAME="E7:14"/>
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">Cl</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <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="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="borsuk_4.html#T20" title="BORSUK_4:th.20">BORSUK_4:20</a></i>;<br/>
<a NAME="E8:14"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">B</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="c5">c</font>,<font color="Maroon" title="c6">d</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="topreal6.html#T80" title="TOPREAL6:th.80">TOPREAL6:80</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:14"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="xxreal_1.html#T95" title="XXREAL_1:th.95">XXREAL_1:95</a></i>;<br/>
<b>hence </b><a NAME="E10:14"/>
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="#E6:14"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="connsp_1.html#D1" title="CONNSP_1:def.1">CONNSP_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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