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<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> ;<br/>

<a NAME="E1:53"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:53_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">a</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:53_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a>, <a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1"/>
<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">a</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:53"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E43</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">b</font>,<font color="Maroon">a</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:53_2"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, </i>;<br/>
<a NAME="E4:53_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">a</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a>, <a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a></i>;<br/>
<b>hence </b><a NAME="E5:53_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:53"/>
<span class="p1">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="isomichi.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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