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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<b>thus </b><a NAME="E2:64"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V3" target="_self">bounded</a> implies (  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V3" target="_self">bounded</a>
 ;<br/>

<a NAME="E2:64_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1"/>
(  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:64"><i><font color="Green">E50</font></i></a>, <a class="ref" href="supinf_2.html#T46" target="_self">Th46</a>, <a class="ref" href="supinf_2.html#T47" target="_self">Th47</a></i>;<br/>


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

<b>assume </b><a NAME="E3:64"/>
(  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<a NAME="E4:64"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:64"><i><font color="Green">E50</font></i></a>, <a class="ref" href="supinf_2.html#T46" target="_self">Th46</a>, <a class="ref" href="supinf_2.html#T47" target="_self">Th47</a></i>;<br/>
<b>hence </b><a NAME="E5:64"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V3" target="_self">bounded</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D13" target="_self">Def13</a></i>;<br/>


</div>
