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<b>let </b><font color="Maroon" title="c1">A</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">A</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a> implies for <font color="Olive" title="b1">r</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>   st  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<b>assume </b><a NAME="E1:134"/><i><font color="Green" title="E77">Z</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</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>   st  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font></span><br/>

<b>let </b><font color="Maroon" title="c2">r</font> be  <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:134"/><i><font color="Green" title="E78">Z4</font></i>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <b>and </b><br/><a NAME="E3:134"/><i><font color="Green" title="E79">Z3</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1"/>
(  ex <font color="Olive" title="b1">y</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>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">y</font> ) or for <font color="Olive" title="b1">y</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>   holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or  not <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">y</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1"><b>suppose </b></a><a NAME="E1:134_1_1"/>
 ex <font color="Olive" title="b1">y</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>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font></span><br/><div class="add"><b>hence </b><a NAME="E2:134_1_1"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E1:134"><i><font color="Green" title="E77">Z</font></i></a>, <a class="txt" href="#E3:134"><i><font color="Green" title="E79">Z3</font></i></a>, <a class="ref" href="xxreal_2.html#T85" target="_self" title="XXREAL_2:th.85">Th85</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_2"><b>suppose </b></a><a NAME="E1:134_1_2"/>
for <font color="Olive" title="b1">y</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>   holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or  not <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font></span><br/><div class="add"><a NAME="E2:134_1_2"/><b>then </b>
<font color="Maroon" title="c2">r</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="xxreal_2.html#D2" target="_self" title="XXREAL_2:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:134_1_2"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E2:134"><i><font color="Green" title="E78">Z4</font></i></a>, <a class="ref" href="xxreal_2.html#D4" target="_self" title="XXREAL_2:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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