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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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"> ( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font> implies ( <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> ) )</span><br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E47">Z</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"> ( <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> &amp; <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> )</span><br/>

<b>thus </b><a NAME="E2:69"/>
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> is <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> is <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E47">Z</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1"/>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E47">Z</font></i></a>, <a class="ref" href="xxreal_2.html#T25" target="_self" title="XXREAL_2:th.25">Th25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xxreal_2.html#D5" target="_self" title="XXREAL_2:def.5">XXREAL_2:def 5</a> <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>

<a NAME="E3:69"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E47">Z</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:69"/>
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E47">Z</font></i></a>, <a class="ref" href="xxreal_2.html#T29" target="_self" title="XXREAL_2:th.29">Th29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xxreal_2.html#D6" target="_self" title="XXREAL_2:def.6">XXREAL_2:def 6</a> <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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