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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c2">Y</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> implies <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> )</span><br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E13">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">Y</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> or <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> )</span><br/>

<b>assume </b><a NAME="E2:22"/>
<font color="Maroon" title="c2">Y</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a></span><br/>

<a NAME="E3:22"/><b>then </b>
 ex <font color="Olive" title="b1">LB</font> being   <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c2">Y</font> st <font color="Olive" title="b1">LB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D12" target="_self" title="SUPINF_1:def.12">Def12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">LB</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:22"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c3">LB</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">LB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 ;<br/>
<a NAME="E6:22"/>
( <font color="Maroon" title="c3">LB</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">LB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:22"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="supinf_1.html#E5:22"><i><font color="Green" title="E14">A2</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="E7:22"/>
<font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a>
 <b>by </b><i><a class="ref" href="supinf_1.html#D12" target="_self" title="SUPINF_1:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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