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<div class="add">

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>thus </b><a NAME="E1:34"/>
<span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/>
<span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="xboole_0.html#R1">meets</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:34_1"/>
( <font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T1" target="_self">Th1</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:34"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#R1">meets</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:34_2"/>
( <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T1" target="_self">Th1</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E6:34_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>


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