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<b>let </b><font color="Maroon" title="c1">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:64"/>
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">D</font> being  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:64"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">D</font> &amp;  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">D</font> )
 <b>by </b><i><a class="ref" href="borsuk_4.html#T82" target="_self" title="BORSUK_4:th.82">Th82</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:64"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">D</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">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <b>by </b><i><a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green" title="E55">A3</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E7:64"/>
( <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E5:64"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E8:64"/><b>then </b>
( <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:64"><i><font color="Green" title="E53">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">p1</font> = <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c6">p2</font> = <font color="Maroon" title="c4">p2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E6:64"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">p1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">p2</font>
;<br/>

<b>thus </b><a NAME="E10:64"/>
<font color="Maroon" title="c5">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">p2</font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E5:64"><i><font color="Green" title="E54">A2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:64"/>
( <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="borsuk_4.html#E5:64"><i><font color="Green" title="E54">A2</font></i></a></i>;<br/>


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