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<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:1_1"/><b>then </b>
<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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E3:1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : (  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:1_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 ;<br/><b>hence </b><a NAME="E5:1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 ;<br/></div>
<b>end;</b></div>

<a NAME="E2:1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E3:1"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E4:1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : (  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:1"/><b>then </b>
<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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>


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