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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<a NAME="E2:66"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c3">S2</font> =  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c2">Y</font>;<br/>
<a NAME="E1:66_1"/>
(  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c2">Y</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c2">Y</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">y</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c2">Y</font> ) )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self" title="SUPINF_1:def.17">Def17</a></i>;<br/>
<a NAME="E2:66_1"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c2">Y</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:66"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="supinf_1.html#T39" target="_self" title="SUPINF_1:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E3:66_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self" title="SUPINF_1:def.17">Def17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font>
 ;<br/>


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