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<b>let </b><font color="Maroon">r2</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">r1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<b>thus </b><a NAME="E2:21"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E2:21_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">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:21_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/>
<a NAME="E5:21_1"/>
<font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:21_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">g2</font> where <font color="Olive">g2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>


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

<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="E3:21"/>
<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">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E4:21"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 ;<br/>
<a NAME="E7:21"/>
<font color="Maroon">g1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g2</font> where <font color="Olive">g2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K2" target="_self">left_closed_halfline</a> <font color="Maroon">r</font>
 ;<br/>


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