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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="supinf_1.html#M3" target="_self">bool_DOMAIN</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:91"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" target="_self">INF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E2:91"/><i><font color="Green">E61</font></i>: 
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" target="_self">INF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="ref" href="supinf_1.html#T118" target="_self">Th118</a></i>;<br/>
<a NAME="E3:91"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of  <a href="supinf_1.html#K9" target="_self">INF</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="ref" href="supinf_1.html#T117" target="_self">Th117</a></i>;<br/>
<a NAME="E4:91"/><b>then </b>
(  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" target="_self">INF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" target="_self">INF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="ref" href="supinf_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E5:91"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" target="_self">INF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T22" target="_self">Th22</a></i>;<br/>


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